platonic-solids-and-prime-numbers-06.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 『正多面体と素数』の計算をしましょう(6)の計算ノート"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import display, Math\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y, X)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('x y X')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#ヘッシアンの行列式を計算\n",
    "def det_hessian(f):\n",
    "    return det(f.hessian()).expand().simplify_full()\n",
    "\n",
    "#二つの関数からヤコビアンを計算\n",
    "def det_jacobian(f,g):\n",
    "    return det(jacobian([f,g],[x,y])).expand().simplify_full()\n",
    "\n",
    "#有理数の分子と分母を素因数分解してlatex形式で返す関数\n",
    "def get_factorized_rational(numerator, denominator):\n",
    "    factors_a = list(factor(numerator))\n",
    "    factors_b = list(factor(denominator))\n",
    "    num = \"\"\n",
    "    if(len(factors_a)==0):\n",
    "        num = \"1\"\n",
    "    else:\n",
    "        for i, t in enumerate(factors_a):\n",
    "            p, k = t\n",
    "            if(k==1):\n",
    "                num += f\"{p}\"\n",
    "            else:\n",
    "                num += f\"{p}^{{{k}}}\"\n",
    "            if(i<len(factors_a)-1):\n",
    "                num += r\" \\!\\cdot\\! \"\n",
    "                \n",
    "    den = \"\"\n",
    "    if(len(factors_b)==0):\n",
    "        den = \"1\"\n",
    "    else:\n",
    "        for i, t in enumerate(factors_b):\n",
    "            p, k = t\n",
    "            if(k==1):\n",
    "                den += f\"{p}\"\n",
    "            else:\n",
    "                den += f\"{p}^{{{k}}}\"\n",
    "            if(i<len(factors_b)-1):\n",
    "                den += r\" \\!\\cdot\\! \"\n",
    "    return \"\\\\frac{{{0}}}{{{1}}}\".format(num, den)\n",
    "\n",
    "# 整数の平方根をいい感じにlatex形式で返す関数\n",
    "def format_factors(squared_factors):\n",
    "    if(len(squared_factors)==0):\n",
    "        return '1'\n",
    "    integer_parts = []\n",
    "    sqrt_parts = []\n",
    "    for i, t in enumerate(squared_factors):\n",
    "        p,k = t\n",
    "        if(k%2==0):\n",
    "            if(k//2==1):\n",
    "                integer_parts.append(f'{p}')\n",
    "            else:\n",
    "                integer_parts.append(f'{p}^{k//2}')\n",
    "        else:\n",
    "            if(k//2==1):\n",
    "                integer_parts.append(f'{p}')\n",
    "            elif(k//2>1):\n",
    "                integer_parts.append(f'{p}^{k//2}')\n",
    "            sqrt_parts.append(f'{p}')\n",
    "    integer_part = ' \\\\!\\\\cdot\\\\! '.join(integer_parts)\n",
    "    if(len(sqrt_parts)==0):\n",
    "        sqrt_part = ''\n",
    "    else:\n",
    "        sqrt_part_inner = ' \\\\!\\\\cdot\\\\! '.join(sqrt_parts)\n",
    "        sqrt_part = f'\\\\sqrt{{{sqrt_part_inner}}}'\n",
    "    return f'{integer_part}{sqrt_part}'\n",
    "\n",
    "#x,yの同次（deg次）多項式fをいい感じに表示する\n",
    "def get_latex_polynomial(f, deg, xy=True):\n",
    "    lines = \"\"\n",
    "    for i in reversed(range(deg+1)):\n",
    "        j = deg-i\n",
    "        if(i<j):break\n",
    "        if(xy):\n",
    "            c=f.coefficient(x^i*y^j)\n",
    "        else:\n",
    "            c=f.coefficient(X^i)\n",
    "        p = numerator(c*c).pyobject()\n",
    "        q = denominator(c*c).pyobject()\n",
    "        if(p>0):\n",
    "            if(xy):\n",
    "                sgn1 = '+' if f.coefficient(x^i*y^j)/f.coefficient(x^j*y^i) == 1 else '-'\n",
    "            else:\n",
    "                sgn1 = '+' if f.coefficient(X^i)/(f.coefficient(X^j) if j > 0 else f(X=0)) == 1 else '-'\n",
    "            if sign(c)>0:\n",
    "                sgn2=\"+\"\n",
    "            else:\n",
    "                sgn2=\"-\"\n",
    "            a = format_factors(p.factor())\n",
    "            b = format_factors(q.factor())\n",
    "            if(a=='1' and b=='1'):\n",
    "                lines += sgn2\n",
    "            elif(b=='1'):\n",
    "                lines += sgn2 + f\"{a}\"\n",
    "            else:\n",
    "                lines += sgn2 + \"\\\\frac{{{0}}}{{{1}}}\".format(a,b)\n",
    "            if(j==0):\n",
    "                if(xy):\n",
    "                    lines += \"\\\\left(x^{{{0}}} {1} y^{{{0}}}\\\\right)\".format(i,sgn1)\n",
    "                else:\n",
    "                    lines += \"\\\\left(X^{{{0}}} {1} 1\\\\right)\".format(i,sgn1)\n",
    "            elif(j==1):\n",
    "                if(xy):\n",
    "                    lines += \"\\\\left(x^{{{0}}}y {1} xy^{{{0}}}\\\\right)\".format(i,sgn1)\n",
    "                else:\n",
    "                    lines += \"\\\\left(X^{{{0}}} {1} X\\\\right)\".format(i,sgn1)\n",
    "            else:\n",
    "                if(xy):\n",
    "                    lines += (\"\\\\left(x^{{{0}}}y^{{{1}}} {2} x^{{{1}}}y^{{{0}}}\\\\right)\".format(i,j,sgn1) if i != j else \"\\\\,x^{{{0}}}y^{{{1}}}\".format(i, j))\n",
    "                else:\n",
    "                    lines += (\"\\\\left(X^{{{0}}} {2} X^{{{1}}}\\\\right)\".format(i,j,sgn1) if i != j else \"\\\\,X^{{{0}}}\".format(i, j))\n",
    "            lines += \"\\n\"\n",
    "    return lines\n",
    "        \n",
    "def print_latex_polynomial(f, deg,xy=True):\n",
    "    print(get_latex_polynomial(f,deg,xy))\n",
    "    \n",
    "def show_latex_polynomial(f,deg,xy=True):\n",
    "    display(Math(get_latex_polynomial(f,deg,xy)))\n",
    "    \n",
    "# 同時多項式の係数の正規化\n",
    "def get_normalized_coefficients(f, deg):\n",
    "    v = []\n",
    "    for i in reversed(range(deg+1)):\n",
    "        j = deg-i\n",
    "        c = f.coefficient(x^i*y^j) * sqrt(factorial(i)*factorial(j))\n",
    "        v.append(c)\n",
    "    u = vector(v).normalized()\n",
    "    w = []\n",
    "    for c in u:\n",
    "        c2 = c^2\n",
    "        p = c2.numerator()\n",
    "        q = c2.denominator()\n",
    "        w.append(sgn(c)*sqrt(p)/sqrt(q))\n",
    "    return vector(w)\n",
    "\n",
    "# 正規化された係数の列を球面調和関数の重ね合わせとして表示\n",
    "def get_as_spherical_harmonics_superposition(f, deg):\n",
    "    v = get_normalized_coefficients(f,deg)\n",
    "    out = \"\"\n",
    "    for i in range(deg+1):\n",
    "        j = deg-i\n",
    "        if(i>j):break\n",
    "        c = v[i]\n",
    "        if(c!=0):\n",
    "            sgn1 = \"+\" if sign(c)>0 else \"-\"\n",
    "            sgn2 = \"+\" if v[i]==v[j] else \"-\"\n",
    "            p = numerator(c*c).pyobject()\n",
    "            q = denominator(c*c).pyobject()\n",
    "            r = get_factorized_rational(p,q)\n",
    "            if(i==j):\n",
    "                out += sgn1 + \"\\\\sqrt{{{0}}}\".format(r) + \"Y^{{{0}}}_{{{1}}}\".format(deg/2, -(i-j)/2)\n",
    "            else:\n",
    "                out += (sgn1 + \"\\\\sqrt{{{0}}}\".format(r)\n",
    "                      + \"\\\\left(Y^{{{0}}}_{{{1}}} {2} Y^{{{0}}}_{{{3}}}\\\\right)\".format(deg/2,-(i-j)/2 , sgn2, (i-j)/2))\n",
    "            out += \"\\n\"\n",
    "    return out\n",
    "\n",
    "def print_as_spherical_harmonics_superposition(f, deg):\n",
    "    print(get_as_spherical_harmonics_superposition(f,deg))\n",
    "    \n",
    "def show_as_spherical_harmonics_superposition(f,deg):\n",
    "    display(Math(get_as_spherical_harmonics_superposition(f,deg)))\n",
    "    \n",
    "def is_prime_power(n):\n",
    "    if(n<=1):\n",
    "        return False\n",
    "    p, k = factor(n)[0]\n",
    "    return p^k == n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28]\n",
      "[1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29]\n"
     ]
    }
   ],
   "source": [
    "A = []\n",
    "B = []\n",
    "for a in range(5):\n",
    "    for b in range(3):\n",
    "        for c in range(2):\n",
    "            d = 6*a+10*b+15*c\n",
    "            if(d<30):\n",
    "                A.append(d)\n",
    "            else:\n",
    "                B.append(d-30)\n",
    "A = sorted(A)\n",
    "B = sorted(B)\n",
    "print(A)\n",
    "print(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# d(n) が 次数2*nの不変式環の次元\n",
    "def dim(i):\n",
    "    if(i%30 in A):\n",
    "        return i//30 + 1\n",
    "    else:\n",
    "        return i//30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 6a + 10b + 15c = i のa>=0, b>=0, c=0,1 なる整数解をタプルの配列として列挙\n",
    "def solutions(i):\n",
    "    s = []\n",
    "    if(i%2==0):\n",
    "        c=0\n",
    "        j = i\n",
    "    else:\n",
    "        c=1\n",
    "        j = i-15\n",
    "    a=0\n",
    "    while((j-6*a)%10!=0):\n",
    "        a += 1;\n",
    "    b = (j-6*a)//10\n",
    "    while(a>=0 and b>=0):\n",
    "        s.append((a,b,c))\n",
    "        a,b=a+5,b-3\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1047 1047\n"
     ]
    }
   ],
   "source": [
    "n = 31415\n",
    "print(len(solutions(n)), dim(n)) #一致する"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# V-座標系"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "V_12(x,y)=x^11*y+11*x^6*y^6-x*y^11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle +\\left(x^{11}y - xy^{11}\\right)\n",
       "+11\\,x^{6}y^{6}\n",
       "$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+\\left(x^{11}y - xy^{11}\\right)\n",
      "+11\\,x^{6}y^{6}\n",
      "\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial(V_12(x,y),12)\n",
    "print_latex_polynomial(V_12(x,y),12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-121 \\, X^{20} + 27588 \\, X^{15} - 59774 \\, X^{10} - 27588 \\, X^{5} - 121</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-121 \\, X^{20} + 27588 \\, X^{15} - 59774 \\, X^{10} - 27588 \\, X^{5} - 121\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-121*X^20 + 27588*X^15 - 59774*X^10 - 27588*X^5 - 121"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "V_20_0(x,y)=det_hessian(V_12(x,y))\n",
    "show(V_20_0(X,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}X^{20} - 228 \\, X^{15} + 494 \\, X^{10} + 228 \\, X^{5} + 1</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}X^{20} - 228 \\, X^{15} + 494 \\, X^{10} + 228 \\, X^{5} + 1\n",
       "\\end{math}"
      ],
      "text/plain": [
       "X^20 - 228*X^15 + 494*X^10 + 228*X^5 + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#最高次の係数を1へ\n",
    "V_20(x,y) = (V_20_0(x,y)/V_20_0(x,y).coefficient(x^20)).expand().simplify_full()\n",
    "show(V_20(X,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle +\\left(x^{20} + y^{20}\\right)\n",
       "-2^2 \\!\\cdot\\! 3 \\!\\cdot\\! 19\\left(x^{15}y^{5} - x^{5}y^{15}\\right)\n",
       "+2 \\!\\cdot\\! 13 \\!\\cdot\\! 19\\,x^{10}y^{10}\n",
       "$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+\\left(x^{20} + y^{20}\\right)\n",
      "-2^2 \\!\\cdot\\! 3 \\!\\cdot\\! 19\\left(x^{15}y^{5} - x^{5}y^{15}\\right)\n",
      "+2 \\!\\cdot\\! 13 \\!\\cdot\\! 19\\,x^{10}y^{10}\n",
      "\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial(V_20(x,y),20)\n",
    "print_latex_polynomial(V_20(x,y),20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-20 \\, x^{30} - 10440 \\, x^{25} y^{5} + 200100 \\, x^{20} y^{10} + 200100 \\, x^{10} y^{20} + 10440 \\, x^{5} y^{25} - 20 \\, y^{30}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-20 \\, x^{30} - 10440 \\, x^{25} y^{5} + 200100 \\, x^{20} y^{10} + 200100 \\, x^{10} y^{20} + 10440 \\, x^{5} y^{25} - 20 \\, y^{30}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "-20*x^30 - 10440*x^25*y^5 + 200100*x^20*y^10 + 200100*x^10*y^20 + 10440*x^5*y^25 - 20*y^30"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "V_30_0(x,y)=det_jacobian(V_12(x,y), V_20(x,y))\n",
    "show(V_30_0(x,y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}X^{30} + 522 \\, X^{25} - 10005 \\, X^{20} - 10005 \\, X^{10} - 522 \\, X^{5} + 1</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}X^{30} + 522 \\, X^{25} - 10005 \\, X^{20} - 10005 \\, X^{10} - 522 \\, X^{5} + 1\n",
       "\\end{math}"
      ],
      "text/plain": [
       "X^30 + 522*X^25 - 10005*X^20 - 10005*X^10 - 522*X^5 + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#最高次の係数を1へ\n",
    "V_30(x,y) = (V_30_0(x,y)/V_30_0(x,y).coefficient(x^30)).expand().simplify_full()\n",
    "show(V_30(X,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle +\\left(x^{30} + y^{30}\\right)\n",
       "+2 \\!\\cdot\\! 3^2 \\!\\cdot\\! 29\\left(x^{25}y^{5} - x^{5}y^{25}\\right)\n",
       "-3 \\!\\cdot\\! 5 \\!\\cdot\\! 23 \\!\\cdot\\! 29\\left(x^{20}y^{10} + x^{10}y^{20}\\right)\n",
       "$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+\\left(x^{30} + y^{30}\\right)\n",
      "+2 \\!\\cdot\\! 3^2 \\!\\cdot\\! 29\\left(x^{25}y^{5} - x^{5}y^{25}\\right)\n",
      "-3 \\!\\cdot\\! 5 \\!\\cdot\\! 23 \\!\\cdot\\! 29\\left(x^{20}y^{10} + x^{10}y^{20}\\right)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial(V_30(x,y),30)\n",
    "print_latex_polynomial(V_30(x,y),30)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "v12(X) = V_12(X,1)\n",
    "v20(X) = V_20(X,1)\n",
    "v30(X) = V_30(X,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1728, 0]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#多項式として除算\n",
    "(v20(X)^3-v30(X)^2).maxima_methods().divide((v12(X)^5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{V_{20}^3-V_{30}^2}{V_{12}^5} = -1728\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "#代表多項式\n",
    "def awesome_polynomial(deg): #def must be an even number\n",
    "    s = solutions(deg//2)\n",
    "    if(len(s)==0):\n",
    "        return 0\n",
    "    f(X) = 0\n",
    "    for i, (a,b,c) in enumerate(s):\n",
    "        g(X) = v12(X)^a * v20(X)^b * v30(X)^c\n",
    "        g(X) = g(X).expand()\n",
    "        #print(g(X))\n",
    "        #print_latex_polynomial((g(x/y)*y^deg).expand(),deg)\n",
    "        #print((g(x/y)*y^deg).expand())\n",
    "        if(i==0):\n",
    "            f(X) = g(X)\n",
    "        else:\n",
    "            k = g(X).degree(X)\n",
    "            #print(-f(X).coefficient(X^k))\n",
    "            #print((-f(X).coefficient(X^k)*(g(x/y)*y^deg)).expand())\n",
    "            f(X) = f(X) - f(X).coefficient(X^k) * g(X)\n",
    "            \n",
    "    return f(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 非零の係数のうち、最高次から2番目。\n",
    "def second_coefficient(f):\n",
    "    return sorted(f.coefficients(), key=lambda x:-x[1])[1][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "#整数nが素数pで割り切れる回数\n",
    "def ord_p(n, p):\n",
    "    if(n==0):\n",
    "        return Infinity\n",
    "    else:\n",
    "        i = 0\n",
    "        while(Mod(n,p)==0):\n",
    "            i += 1\n",
    "            n = n/p\n",
    "        return i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "12\t11\t1\tTrue\t11\n",
      "20\t19\t1\tTrue\t-228\n",
      "24\t23\t1\tFalse\t22\n",
      "30\t29\t1\tTrue\t522\n",
      "32\t31\t1\tTrue\t-217\n",
      "42\t41\t1\tTrue\t533\n",
      "44\t43\t1\tFalse\t-206\n",
      "48\t47\t1\tFalse\t44\n",
      "50\t7\t2\tTrue\t294\n",
      "54\t53\t1\tFalse\t544\n",
      "60\t59\t1\tTrue\t195054\n",
      "62\t61\t1\tTrue\t305\n",
      "68\t67\t1\tFalse\t-184\n",
      "72\t71\t1\tTrue\t194327\n",
      "74\t73\t1\tFalse\t316\n",
      "80\t79\t1\tTrue\t156104\n",
      "82\t3\t4\tFalse\t77\n",
      "84\t83\t1\tFalse\t193479\n",
      "90\t89\t1\tTrue\t-116145\n",
      "98\t97\t1\tFalse\t338\n",
      "102\t101\t1\tTrue\t-122614\n",
      "104\t103\t1\tFalse\t159545\n",
      "108\t107\t1\tFalse\t191420\n",
      "110\t109\t1\tTrue\t-36079\n",
      "114\t113\t1\tFalse\t-129204\n",
      "122\t11\t2\tTrue\t-40040\n",
      "126\t5\t3\tFalse\t-135915\n",
      "128\t127\t1\tFalse\t162502\n",
      "132\t131\t1\tTrue\t-68395886\n",
      "138\t137\t1\tFalse\t-142747\n",
      "140\t139\t1\tTrue\t-28913668\n",
      "150\t149\t1\tTrue\t18148200\n",
      "152\t151\t1\tTrue\t-30728500\n",
      "158\t157\t1\tFalse\t-52649\n",
      "164\t163\t1\tFalse\t-32554926\n",
      "168\t167\t1\tFalse\t-74527176\n",
      "170\t13\t2\tTrue\t4931420\n",
      "174\t173\t1\tFalse\t21677670\n",
      "180\t179\t1\tTrue\t13492680795\n",
      "182\t181\t1\tTrue\t5610095\n",
      "192\t191\t1\tTrue\t14356548043\n",
      "194\t193\t1\tFalse\t6340338\n",
      "198\t197\t1\tFalse\t25529088\n",
      "200\t199\t1\tTrue\t4724847050\n",
      "212\t211\t1\tTrue\t5164465540\n",
      "224\t223\t1\tFalse\t5624593221\n",
      "228\t227\t1\tFalse\t17076532596\n",
      "230\t229\t1\tTrue\t-708584395\n",
      "234\t233\t1\tFalse\t-3445727520\n",
      "240\t239\t1\tTrue\t-2398869114096\n",
      "242\t241\t1\tTrue\t-816510410\n",
      "244\t3\t5\tFalse\t1715817158\n",
      "252\t251\t1\tTrue\t-2607891398070\n",
      "258\t257\t1\tFalse\t-4225795245\n",
      "264\t263\t1\tFalse\t-2827795269068\n",
      "270\t269\t1\tTrue\t423167756922\n",
      "272\t271\t1\tTrue\t-834643499823\n",
      "278\t277\t1\tFalse\t-1209486876\n",
      "282\t281\t1\tTrue\t479477438805\n",
      "284\t283\t1\tFalse\t-925265824770\n",
      "290\t17\t2\tTrue\t104781001780\n",
      "294\t293\t1\tFalse\t541160889280\n",
      "308\t307\t1\tFalse\t-1125996877184\n",
      "312\t311\t1\tTrue\t447184277594197\n",
      "314\t313\t1\tFalse\t140633113488\n",
      "318\t317\t1\tFalse\t681998590988\n",
      "332\t331\t1\tTrue\t133157198898746\n",
      "338\t337\t1\tFalse\t185292548566\n",
      "344\t7\t3\tFalse\t149439501382416\n",
      "348\t347\t1\tFalse\t592190531112166\n",
      "350\t349\t1\tTrue\t-15785874597479\n",
      "354\t353\t1\tFalse\t-84858744582729\n",
      "360\t359\t1\tTrue\t-66840871681048104\n",
      "362\t19\t2\tTrue\t-18430698297120\n",
      "368\t367\t1\tFalse\t186464262299406\n",
      "374\t373\t1\tFalse\t-21425419944929\n",
      "380\t379\t1\tTrue\t-18622642220906256\n",
      "384\t383\t1\tFalse\t-83068997146837006\n",
      "390\t389\t1\tTrue\t10094175129981540\n",
      "398\t397\t1\tFalse\t-28606280449376\n",
      "402\t401\t1\tTrue\t11618079526069128\n",
      "410\t409\t1\tTrue\t2409535633659648\n",
      "420\t419\t1\tTrue\t10899302142379889700\n",
      "422\t421\t1\tTrue\t2823971578022675\n",
      "432\t431\t1\tTrue\t12271475132531877845\n",
      "434\t433\t1\tFalse\t3296937733354590\n",
      "440\t439\t1\tTrue\t2936685655569902885\n",
      "444\t443\t1\tFalse\t13781949709201809135\n",
      "450\t449\t1\tTrue\t-1573095085680484800\n",
      "458\t457\t1\tFalse\t4445309473284512\n",
      "462\t461\t1\tTrue\t-1819683549770405841\n",
      "464\t463\t1\tFalse\t3829153759562391403\n",
      "468\t467\t1\tFalse\t17260978727420295028\n",
      "480\t479\t1\tTrue\t-1766338539213450343520\n",
      "488\t487\t1\tFalse\t4937483203223127766\n",
      "492\t491\t1\tTrue\t-2002251263130369839133\n",
      "500\t499\t1\tTrue\t-464052195253348513000\n",
      "504\t503\t1\tFalse\t-2264291276215480419250\n",
      "510\t509\t1\tTrue\t246352691833237143830\n",
      "522\t521\t1\tTrue\t286093102294335607599\n",
      "524\t523\t1\tFalse\t-611537963402679691014\n",
      "530\t23\t2\tTrue\t57664391382790722340\n",
      "542\t541\t1\tTrue\t67929251018864698470\n",
      "548\t547\t1\tFalse\t-797554927770640524636\n",
      "558\t557\t1\tFalse\t440758252426438927450\n",
      "564\t563\t1\tFalse\t369821473887493716535865\n",
      "570\t569\t1\tTrue\t-38740042331942491459035\n",
      "572\t571\t1\tTrue\t84835540357844344289415\n",
      "578\t577\t1\tFalse\t109019414844211345110\n",
      "588\t587\t1\tFalse\t475218319272787656834160\n",
      "594\t593\t1\tFalse\t-52442079678808401403640\n",
      "600\t599\t1\tTrue\t-46005882852976266379473600\n",
      "602\t601\t1\tTrue\t-10629558332421527688540\n",
      "608\t607\t1\tFalse\t128578423799083333597752\n",
      "614\t613\t1\tFalse\t-12508207935566100348459\n",
      "618\t617\t1\tFalse\t-70277794776861932591584\n",
      "620\t619\t1\tTrue\t-11664490234408945766161252\n",
      "626\t5\t4\tFalse\t-14677999646952390288309\n",
      "632\t631\t1\tTrue\t-13511235648591982629241553\n",
      "642\t641\t1\tTrue\t7141027704452508733943576\n",
      "644\t643\t1\tFalse\t-15613573005579365604774512\n",
      "648\t647\t1\tFalse\t-78080333835825534863016512\n",
      "654\t653\t1\tFalse\t8320793750396364670465068\n",
      "660\t659\t1\tTrue\t7412477615946471007131759798\n",
      "662\t661\t1\tTrue\t1670794226722568532564570\n",
      "674\t673\t1\tFalse\t1969441929782752406403404\n",
      "678\t677\t1\tFalse\t11218573313220325818431904\n",
      "684\t683\t1\tFalse\t9771543468807440987427211286\n",
      "692\t691\t1\tTrue\t2155037845999660250735164500\n",
      "702\t701\t1\tTrue\t-1132851908306349431284326425\n",
      "710\t709\t1\tTrue\t-222765564279437691268451800\n",
      "720\t719\t1\tTrue\t-1193992231086088963699063608000\n",
      "728\t727\t1\tFalse\t3334678373523333138504742475\n",
      "730\t3\t6\tFalse\t-48950956575227128555733600\n",
      "734\t733\t1\tFalse\t-311170414198271384114958600\n",
      "740\t739\t1\tTrue\t-295598629588161813276760604900\n",
      "744\t743\t1\tFalse\t-1584337752243836711428686711150\n",
      "752\t751\t1\tTrue\t-344225541080439606854726625000\n",
      "758\t757\t1\tFalse\t-430523181272288571465099101\n",
      "762\t761\t1\tTrue\t180131922671414189271996551868\n",
      "770\t769\t1\tTrue\t35221385611282839053363129390\n",
      "774\t773\t1\tFalse\t210803788556357549260737840640\n",
      "788\t787\t1\tFalse\t-536973034213688727053245953092\n",
      "798\t797\t1\tFalse\t286929559810608055899610429476\n",
      "810\t809\t1\tTrue\t-24433905259171659220311786200730\n",
      "812\t811\t1\tTrue\t55059127616937253619059785079110\n",
      "822\t821\t1\tTrue\t-28701134186773110225371322913344\n",
      "824\t823\t1\tFalse\t64125558726861117197934779087769\n",
      "828\t827\t1\tFalse\t340083536825583752705993319334644\n",
      "830\t829\t1\tTrue\t-5584851581432485330945178826925\n",
      "840\t839\t1\tTrue\t-30992329023844330416433975458784680\n",
      "842\t29\t2\tTrue\t-6621647025461261626823518940400\n",
      "854\t853\t1\tFalse\t-7834083235841866620035582701450\n",
      "858\t857\t1\tFalse\t-45969965028451886251451041242075\n",
      "860\t859\t1\tTrue\t-7542800491872460522993427486232460\n",
      "864\t863\t1\tFalse\t-41561195005348884298710643986401210\n",
      "878\t877\t1\tFalse\t-10897550968251981381434760852035\n",
      "882\t881\t1\tTrue\t4581427003541767325732534711957720\n",
      "884\t883\t1\tFalse\t-10290021123615998901333388121302680\n",
      "888\t887\t1\tFalse\t-55358424427207555350297646054227680\n",
      "908\t907\t1\tFalse\t-13936561409257484219235561861101736\n",
      "912\t911\t1\tTrue\t5802555965464257074417197810818483438\n",
      "920\t919\t1\tTrue\t1207560913287110185043953753981506441\n",
      "930\t929\t1\tTrue\t-621980566450161991718813230425976764\n",
      "938\t937\t1\tFalse\t1738968522892010666161356011915088\n",
      "942\t941\t1\tTrue\t-732508195910899589869977211041554954\n",
      "948\t947\t1\tFalse\t9003651824685102100850748782449812408\n",
      "954\t953\t1\tFalse\t-861129170841468877938317711731265472\n",
      "962\t31\t2\tTrue\t-167921525651633730755743114502900450\n",
      "968\t967\t1\tFalse\t2246715134998035594303073446236310210\n",
      "972\t971\t1\tTrue\t-937524853965474675564241363024726174380\n",
      "978\t977\t1\tFalse\t-1183883414542177756774941456505218090\n",
      "984\t983\t1\tFalse\t-1089242812664817846577169706375448973360\n",
      "992\t991\t1\tTrue\t-226988297296954801973741620257412091550\n",
      "998\t997\t1\tFalse\t-277965324633336616164021115016853115\n",
      "1010\t1009\t1\tTrue\t22571816475858432283419337197410893470\n",
      "1014\t1013\t1\tFalse\t138046157547894389299186471848689274720\n",
      "1020\t1019\t1\tTrue\t129992688425332623556195797780831190334150\n",
      "1022\t1021\t1\tTrue\t26817509841275416292897502695360591225\n",
      "1032\t1031\t1\tTrue\t151520273276886367318346314045083187571540\n",
      "1034\t1033\t1\tFalse\t31804498219459391602060725061473231980\n",
      "1040\t1039\t1\tTrue\t31067909913890522341453013091384805839920\n",
      "1050\t1049\t1\tTrue\t-15935733038770489605524999498356451715425\n",
      "1052\t1051\t1\tTrue\t36476213406769574208704337720428854853625\n",
      "1062\t1061\t1\tTrue\t-18805711410546892187764581962612428589764\n",
      "1064\t1063\t1\tFalse\t42757796400539347090417828399415160994182\n",
      "1070\t1069\t1\tTrue\t-3608697669874351696101296005261335540310\n",
      "1088\t1087\t1\tFalse\t58479577234643258263288682327190417890628\n",
      "1092\t1091\t1\tTrue\t-24496494126835970079524067374082380945372639\n",
      "1094\t1093\t1\tFalse\t-5090716485178668307778656216750789108464\n",
      "1098\t1097\t1\tFalse\t-30610313517254502314313402657807147339714\n",
      "1104\t1103\t1\tFalse\t-28554704256265686862858371992787192882957750\n",
      "1110\t1109\t1\tTrue\t2555937464165644916005186056122254931557200\n",
      "1118\t1117\t1\tFalse\t-7133423886813601394888091371333148689570\n",
      "1124\t1123\t1\tFalse\t-6885874423029873422480356725833190278302310\n",
      "1130\t1129\t1\tTrue\t577816612751592526785929323751691383437400\n",
      "1152\t1151\t1\tTrue\t3961818785423782056618304171930577928129351785\n",
      "1154\t1153\t1\tFalse\t815966264872806310468160038164348862294700\n",
      "1164\t1163\t1\tFalse\t4624656599659394577037466454006361498962277370\n",
      "1172\t1171\t1\tTrue\t944591811946866921900966635855993775090904420\n",
      "1182\t1181\t1\tTrue\t-485140613092031389627573620642153045053669035\n",
      "1188\t1187\t1\tFalse\t6276010375049518531579242531895016240379666940\n",
      "1194\t1193\t1\tFalse\t-572598269768179609683145511120643831141018960\n",
      "1202\t1201\t1\tTrue\t-110230210175568582761855712989023602562277160\n",
      "1214\t1213\t1\tFalse\t-130951261383652505950823361568438855123227396\n",
      "1218\t1217\t1\tFalse\t-794201922426950644958742610238430703228394271\n",
      "1224\t1223\t1\tFalse\t-749183831942768490980411015769374139297593823738\n",
      "1230\t1229\t1\tTrue\t65978377240737478533203657550542747931048313482\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1232\t1231\t1\tTrue\t-152199967751824437366272677726785244922748803007\n",
      "1238\t1237\t1\tFalse\t-183973981146877766375242211328830690862158370\n",
      "1250\t1249\t1\tTrue\t14872335977419832498007033310299483386244338280\n",
      "1260\t1259\t1\tTrue\t88545907209215863675006932448160972392670344719180\n",
      "1278\t1277\t1\tFalse\t128075206628429865801138172819949777175852067060\n",
      "1280\t1279\t1\tTrue\t20822348341218047970897155140835876971574979469440\n",
      "1284\t1283\t1\tFalse\t121398818713138038482713531161063851243225513874440\n",
      "1290\t1289\t1\tTrue\t-10616682801076541291613991662788154300975633031050\n",
      "1292\t1291\t1\tTrue\t24542533785441785928054621293929857557591624298150\n",
      "1298\t1297\t1\tFalse\t29591924352761074762859639106096879577421700370\n",
      "1302\t1301\t1\tTrue\t-12565940993877216800846427354413839631650161517860\n",
      "1304\t1303\t1\tFalse\t28888612637284925170013489215087194890054959882900\n",
      "1308\t1307\t1\tFalse\t165602097604134580270554310664990647144053738707900\n",
      "1320\t1319\t1\tTrue\t-14322341237880207071941785639226894058451941230558800\n",
      "1322\t1321\t1\tTrue\t-2845969287046341207447960636535739983241456493200\n",
      "1328\t1327\t1\tFalse\t39868660594383736030383724996454647667002186579660\n",
      "1332\t11\t3\tTrue\t-16798001310344416588618982042557454404989338606632630\n"
     ]
    }
   ],
   "source": [
    "#最初の予想の検証。整数係数で計算。\n",
    "N=1332\n",
    "T = []\n",
    "S = []\n",
    "for deg in range(0,N+1,2):\n",
    "    if(dim(deg//2)==0):\n",
    "        continue\n",
    "    if(is_prime_power(deg-1)):\n",
    "        p,k = factor(deg-1)[0]\n",
    "    else:\n",
    "        continue\n",
    "    f = awesome_polynomial(deg)\n",
    "    s = second_coefficient(f)\n",
    "    S.append([deg,s])\n",
    "    ok = all([ord_p(c,p)+ord_p(i,p)+ord_p(deg-i,p) >= k for [c,i] in f.coefficients()])\n",
    "    if(ok):\n",
    "        T.append(deg-1)\n",
    "    print(f\"{deg}\\t{p}\\t{k}\\t{ok}\\t{s}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[11, 19, 29, 31, 41]"
      ]
     },
     "execution_count": 221,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[9, 11, 19, 29, 31]"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U = list(filter(lambda q: is_prime_power(q) and (q%10==1 or q%10==9), range(1, N+1, 2)))\n",
    "U[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{9, 81, 729}"
      ]
     },
     "execution_count": 223,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(U)-set(T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "set()"
      ]
     },
     "execution_count": 224,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(T)-set(U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 3\n",
    "k = 6\n",
    "deg = p^k+1\n",
    "f = awesome_polynomial(deg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\t0\t1\n",
      "310\t5\t-1758886940740495904692316758754884475657072172633801816323789698601764935422231604493970161853603400\n",
      "325\t2\t-65089682511371964392165499442245466171324834822680121472172834473419557541833199348713699725001300\n",
      "405\t2\t65089682511371964392165499442245466171324834822680121472172834473419557541833199348713699725001300\n",
      "420\t5\t-1758886940740495904692316758754884475657072172633801816323789698601764935422231604493970161853603400\n",
      "730\t0\t1\n"
     ]
    }
   ],
   "source": [
    "for [c,i] in f.coefficients():\n",
    "    if(ord_p(c,p)+ord_p(i,p)+ord_p(deg-i,p) >= k):\n",
    "        print(f\"{i}\\t{ord_p(c,p)}\\t{c}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-7238217863129612776511591599814339405996181780386015705036171599184217841243751458822922476763800"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-1758886940740495904692316758754884475657072172633801816323789698601764935422231604493970161853603400/3^5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7232186945707996043573944382471718463480537202520013496908092719268839726870355483190411080555700"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "65089682511371964392165499442245466171324834822680121472172834473419557541833199348713699725001300/3^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "#代表多項式をmodで計算\n",
    "def awesome_polynomial_mod(deg, q): #def must be an even number\n",
    "    R = Integers(q)['X']\n",
    "    w12=R(v12(X))\n",
    "    w20=R(v20(X))\n",
    "    w30=R(v30(X))\n",
    "    s = solutions(deg//2)\n",
    "    if(len(s)==0):\n",
    "        return 0\n",
    "    f = R(0)\n",
    "    for i, (a,b,c) in enumerate(s):\n",
    "        g = w12^a * w20^b * w30^c\n",
    "        if(i==0):\n",
    "            f = g\n",
    "        else:\n",
    "            k = g.degree()\n",
    "            f = f - f.list()[k] * g\n",
    "\n",
    "    return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^121 + 11*X^111 + 66*X^66 + 66*X^56 + 110*X^11 + 120*X"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "awesome_polynomial_mod(122,121)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^121 + 10*X"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "awesome_polynomial_mod(122,11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^124 + 69*X^114 + 27*X^109 + 78*X^104 + 112*X^99 + 89*X^94 + 20*X^89 + 39*X^84 + 29*X^79 + 33*X^74 + 56*X^69 + 70*X^64 + 71*X^59 + 33*X^54 + 98*X^49 + 39*X^44 + 107*X^39 + 89*X^34 + 15*X^29 + 78*X^24 + 100*X^19 + 69*X^14 + X^4"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "awesome_polynomial_mod(128,127)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "12\t11\t1\tTrue\n",
      "20\t19\t1\tTrue\n",
      "24\t23\t1\tFalse\n",
      "30\t29\t1\tTrue\n",
      "32\t31\t1\tTrue\n",
      "42\t41\t1\tTrue\n",
      "44\t43\t1\tFalse\n",
      "48\t47\t1\tFalse\n",
      "50\t7\t2\tTrue\n",
      "54\t53\t1\tFalse\n",
      "60\t59\t1\tTrue\n",
      "62\t61\t1\tTrue\n",
      "68\t67\t1\tFalse\n",
      "72\t71\t1\tTrue\n",
      "74\t73\t1\tFalse\n",
      "80\t79\t1\tTrue\n",
      "82\t3\t4\tFalse\n",
      "84\t83\t1\tFalse\n",
      "90\t89\t1\tTrue\n",
      "98\t97\t1\tFalse\n",
      "102\t101\t1\tTrue\n",
      "104\t103\t1\tFalse\n",
      "108\t107\t1\tFalse\n",
      "110\t109\t1\tTrue\n",
      "114\t113\t1\tFalse\n",
      "122\t11\t2\tTrue\n",
      "126\t5\t3\tFalse\n",
      "128\t127\t1\tFalse\n",
      "132\t131\t1\tTrue\n",
      "138\t137\t1\tFalse\n",
      "140\t139\t1\tTrue\n",
      "150\t149\t1\tTrue\n",
      "152\t151\t1\tTrue\n",
      "158\t157\t1\tFalse\n",
      "164\t163\t1\tFalse\n",
      "168\t167\t1\tFalse\n",
      "170\t13\t2\tTrue\n",
      "174\t173\t1\tFalse\n",
      "180\t179\t1\tTrue\n",
      "182\t181\t1\tTrue\n",
      "192\t191\t1\tTrue\n",
      "194\t193\t1\tFalse\n",
      "198\t197\t1\tFalse\n",
      "200\t199\t1\tTrue\n",
      "212\t211\t1\tTrue\n",
      "224\t223\t1\tFalse\n",
      "228\t227\t1\tFalse\n",
      "230\t229\t1\tTrue\n",
      "234\t233\t1\tFalse\n",
      "240\t239\t1\tTrue\n",
      "242\t241\t1\tTrue\n",
      "244\t3\t5\tFalse\n",
      "252\t251\t1\tTrue\n",
      "258\t257\t1\tFalse\n",
      "264\t263\t1\tFalse\n",
      "270\t269\t1\tTrue\n",
      "272\t271\t1\tTrue\n",
      "278\t277\t1\tFalse\n",
      "282\t281\t1\tTrue\n",
      "284\t283\t1\tFalse\n",
      "290\t17\t2\tTrue\n",
      "294\t293\t1\tFalse\n",
      "308\t307\t1\tFalse\n",
      "312\t311\t1\tTrue\n",
      "314\t313\t1\tFalse\n",
      "318\t317\t1\tFalse\n",
      "332\t331\t1\tTrue\n",
      "338\t337\t1\tFalse\n",
      "344\t7\t3\tFalse\n",
      "348\t347\t1\tFalse\n",
      "350\t349\t1\tTrue\n",
      "354\t353\t1\tFalse\n",
      "360\t359\t1\tTrue\n",
      "362\t19\t2\tTrue\n",
      "368\t367\t1\tFalse\n",
      "374\t373\t1\tFalse\n",
      "380\t379\t1\tTrue\n",
      "384\t383\t1\tFalse\n",
      "390\t389\t1\tTrue\n",
      "398\t397\t1\tFalse\n",
      "402\t401\t1\tTrue\n",
      "410\t409\t1\tTrue\n",
      "420\t419\t1\tTrue\n",
      "422\t421\t1\tTrue\n",
      "432\t431\t1\tTrue\n",
      "434\t433\t1\tFalse\n",
      "440\t439\t1\tTrue\n",
      "444\t443\t1\tFalse\n",
      "450\t449\t1\tTrue\n",
      "458\t457\t1\tFalse\n",
      "462\t461\t1\tTrue\n",
      "464\t463\t1\tFalse\n",
      "468\t467\t1\tFalse\n",
      "480\t479\t1\tTrue\n",
      "488\t487\t1\tFalse\n",
      "492\t491\t1\tTrue\n",
      "500\t499\t1\tTrue\n",
      "504\t503\t1\tFalse\n",
      "510\t509\t1\tTrue\n",
      "522\t521\t1\tTrue\n",
      "524\t523\t1\tFalse\n",
      "530\t23\t2\tTrue\n",
      "542\t541\t1\tTrue\n",
      "548\t547\t1\tFalse\n",
      "558\t557\t1\tFalse\n",
      "564\t563\t1\tFalse\n",
      "570\t569\t1\tTrue\n",
      "572\t571\t1\tTrue\n",
      "578\t577\t1\tFalse\n",
      "588\t587\t1\tFalse\n",
      "594\t593\t1\tFalse\n",
      "600\t599\t1\tTrue\n",
      "602\t601\t1\tTrue\n",
      "608\t607\t1\tFalse\n",
      "614\t613\t1\tFalse\n",
      "618\t617\t1\tFalse\n",
      "620\t619\t1\tTrue\n",
      "626\t5\t4\tFalse\n",
      "632\t631\t1\tTrue\n",
      "642\t641\t1\tTrue\n",
      "644\t643\t1\tFalse\n",
      "648\t647\t1\tFalse\n",
      "654\t653\t1\tFalse\n",
      "660\t659\t1\tTrue\n",
      "662\t661\t1\tTrue\n",
      "674\t673\t1\tFalse\n",
      "678\t677\t1\tFalse\n",
      "684\t683\t1\tFalse\n",
      "692\t691\t1\tTrue\n",
      "702\t701\t1\tTrue\n",
      "710\t709\t1\tTrue\n",
      "720\t719\t1\tTrue\n",
      "728\t727\t1\tFalse\n",
      "730\t3\t6\tFalse\n",
      "734\t733\t1\tFalse\n",
      "740\t739\t1\tTrue\n",
      "744\t743\t1\tFalse\n",
      "752\t751\t1\tTrue\n",
      "758\t757\t1\tFalse\n",
      "762\t761\t1\tTrue\n",
      "770\t769\t1\tTrue\n",
      "774\t773\t1\tFalse\n",
      "788\t787\t1\tFalse\n",
      "798\t797\t1\tFalse\n",
      "810\t809\t1\tTrue\n",
      "812\t811\t1\tTrue\n",
      "822\t821\t1\tTrue\n",
      "824\t823\t1\tFalse\n",
      "828\t827\t1\tFalse\n",
      "830\t829\t1\tTrue\n",
      "840\t839\t1\tTrue\n",
      "842\t29\t2\tTrue\n",
      "854\t853\t1\tFalse\n",
      "858\t857\t1\tFalse\n",
      "860\t859\t1\tTrue\n",
      "864\t863\t1\tFalse\n",
      "878\t877\t1\tFalse\n",
      "882\t881\t1\tTrue\n",
      "884\t883\t1\tFalse\n",
      "888\t887\t1\tFalse\n",
      "908\t907\t1\tFalse\n",
      "912\t911\t1\tTrue\n",
      "920\t919\t1\tTrue\n",
      "930\t929\t1\tTrue\n",
      "938\t937\t1\tFalse\n",
      "942\t941\t1\tTrue\n",
      "948\t947\t1\tFalse\n",
      "954\t953\t1\tFalse\n",
      "962\t31\t2\tTrue\n",
      "968\t967\t1\tFalse\n",
      "972\t971\t1\tTrue\n",
      "978\t977\t1\tFalse\n",
      "984\t983\t1\tFalse\n",
      "992\t991\t1\tTrue\n",
      "998\t997\t1\tFalse\n",
      "1010\t1009\t1\tTrue\n",
      "1014\t1013\t1\tFalse\n",
      "1020\t1019\t1\tTrue\n",
      "1022\t1021\t1\tTrue\n",
      "1032\t1031\t1\tTrue\n",
      "1034\t1033\t1\tFalse\n",
      "1040\t1039\t1\tTrue\n",
      "1050\t1049\t1\tTrue\n",
      "1052\t1051\t1\tTrue\n",
      "1062\t1061\t1\tTrue\n",
      "1064\t1063\t1\tFalse\n",
      "1070\t1069\t1\tTrue\n",
      "1088\t1087\t1\tFalse\n",
      "1092\t1091\t1\tTrue\n",
      "1094\t1093\t1\tFalse\n",
      "1098\t1097\t1\tFalse\n",
      "1104\t1103\t1\tFalse\n",
      "1110\t1109\t1\tTrue\n",
      "1118\t1117\t1\tFalse\n",
      "1124\t1123\t1\tFalse\n",
      "1130\t1129\t1\tTrue\n",
      "1152\t1151\t1\tTrue\n",
      "1154\t1153\t1\tFalse\n",
      "1164\t1163\t1\tFalse\n",
      "1172\t1171\t1\tTrue\n",
      "1182\t1181\t1\tTrue\n",
      "1188\t1187\t1\tFalse\n",
      "1194\t1193\t1\tFalse\n",
      "1202\t1201\t1\tTrue\n",
      "1214\t1213\t1\tFalse\n",
      "1218\t1217\t1\tFalse\n",
      "1224\t1223\t1\tFalse\n",
      "1230\t1229\t1\tTrue\n",
      "1232\t1231\t1\tTrue\n",
      "1238\t1237\t1\tFalse\n",
      "1250\t1249\t1\tTrue\n",
      "1260\t1259\t1\tTrue\n",
      "1278\t1277\t1\tFalse\n",
      "1280\t1279\t1\tTrue\n",
      "1284\t1283\t1\tFalse\n",
      "1290\t1289\t1\tTrue\n",
      "1292\t1291\t1\tTrue\n",
      "1298\t1297\t1\tFalse\n",
      "1302\t1301\t1\tTrue\n",
      "1304\t1303\t1\tFalse\n",
      "1308\t1307\t1\tFalse\n",
      "1320\t1319\t1\tTrue\n",
      "1322\t1321\t1\tTrue\n",
      "1328\t1327\t1\tFalse\n",
      "1332\t11\t3\tTrue\n",
      "1362\t1361\t1\tTrue\n",
      "1368\t1367\t1\tFalse\n",
      "1370\t37\t2\tTrue\n",
      "1374\t1373\t1\tFalse\n",
      "1382\t1381\t1\tTrue\n",
      "1400\t1399\t1\tTrue\n",
      "1410\t1409\t1\tTrue\n",
      "1424\t1423\t1\tFalse\n",
      "1428\t1427\t1\tFalse\n",
      "1430\t1429\t1\tTrue\n",
      "1434\t1433\t1\tFalse\n",
      "1440\t1439\t1\tTrue\n",
      "1448\t1447\t1\tFalse\n",
      "1452\t1451\t1\tTrue\n",
      "1454\t1453\t1\tFalse\n",
      "1460\t1459\t1\tTrue\n",
      "1472\t1471\t1\tTrue\n",
      "1482\t1481\t1\tTrue\n",
      "1484\t1483\t1\tFalse\n",
      "1488\t1487\t1\tFalse\n",
      "1490\t1489\t1\tTrue\n",
      "1494\t1493\t1\tFalse\n",
      "1500\t1499\t1\tTrue\n",
      "1512\t1511\t1\tTrue\n",
      "1524\t1523\t1\tFalse\n",
      "1532\t1531\t1\tTrue\n",
      "1544\t1543\t1\tFalse\n",
      "1550\t1549\t1\tTrue\n",
      "1554\t1553\t1\tFalse\n",
      "1560\t1559\t1\tTrue\n",
      "1568\t1567\t1\tFalse\n",
      "1572\t1571\t1\tTrue\n",
      "1580\t1579\t1\tTrue\n",
      "1584\t1583\t1\tFalse\n",
      "1598\t1597\t1\tFalse\n",
      "1602\t1601\t1\tTrue\n",
      "1608\t1607\t1\tFalse\n",
      "1610\t1609\t1\tTrue\n",
      "1614\t1613\t1\tFalse\n",
      "1620\t1619\t1\tTrue\n",
      "1622\t1621\t1\tTrue\n",
      "1628\t1627\t1\tFalse\n",
      "1638\t1637\t1\tFalse\n",
      "1658\t1657\t1\tFalse\n",
      "1664\t1663\t1\tFalse\n",
      "1668\t1667\t1\tFalse\n",
      "1670\t1669\t1\tTrue\n",
      "1682\t41\t2\tTrue\n",
      "1694\t1693\t1\tFalse\n",
      "1698\t1697\t1\tFalse\n",
      "1700\t1699\t1\tTrue\n",
      "1710\t1709\t1\tTrue\n",
      "1722\t1721\t1\tTrue\n",
      "1724\t1723\t1\tFalse\n",
      "1734\t1733\t1\tFalse\n",
      "1742\t1741\t1\tTrue\n",
      "1748\t1747\t1\tFalse\n",
      "1754\t1753\t1\tFalse\n",
      "1760\t1759\t1\tTrue\n",
      "1778\t1777\t1\tFalse\n",
      "1784\t1783\t1\tFalse\n",
      "1788\t1787\t1\tFalse\n",
      "1790\t1789\t1\tTrue\n",
      "1802\t1801\t1\tTrue\n",
      "1812\t1811\t1\tTrue\n",
      "1824\t1823\t1\tFalse\n",
      "1832\t1831\t1\tTrue\n",
      "1848\t1847\t1\tFalse\n",
      "1850\t43\t2\tTrue\n",
      "1862\t1861\t1\tTrue\n",
      "1868\t1867\t1\tFalse\n",
      "1872\t1871\t1\tTrue\n",
      "1874\t1873\t1\tFalse\n",
      "1878\t1877\t1\tFalse\n",
      "1880\t1879\t1\tTrue\n",
      "1890\t1889\t1\tTrue\n",
      "1902\t1901\t1\tTrue\n",
      "1908\t1907\t1\tFalse\n",
      "1914\t1913\t1\tFalse\n",
      "1932\t1931\t1\tTrue\n",
      "1934\t1933\t1\tFalse\n",
      "1950\t1949\t1\tTrue\n",
      "1952\t1951\t1\tTrue\n",
      "1974\t1973\t1\tFalse\n",
      "1980\t1979\t1\tTrue\n",
      "1988\t1987\t1\tFalse\n",
      "1994\t1993\t1\tFalse\n",
      "1998\t1997\t1\tFalse\n",
      "2000\t1999\t1\tTrue\n",
      "2004\t2003\t1\tFalse\n",
      "2012\t2011\t1\tTrue\n",
      "2018\t2017\t1\tFalse\n",
      "2028\t2027\t1\tFalse\n",
      "2030\t2029\t1\tTrue\n",
      "2040\t2039\t1\tTrue\n",
      "2054\t2053\t1\tFalse\n",
      "2064\t2063\t1\tFalse\n",
      "2070\t2069\t1\tTrue\n",
      "2082\t2081\t1\tTrue\n",
      "2084\t2083\t1\tFalse\n",
      "2088\t2087\t1\tFalse\n",
      "2090\t2089\t1\tTrue\n",
      "2100\t2099\t1\tTrue\n",
      "2112\t2111\t1\tTrue\n",
      "2114\t2113\t1\tFalse\n",
      "2130\t2129\t1\tTrue\n",
      "2132\t2131\t1\tTrue\n",
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "9344\t9343\t1\tFalse\n",
      "9350\t9349\t1\tTrue\n",
      "9372\t9371\t1\tTrue\n",
      "9378\t9377\t1\tFalse\n",
      "9392\t9391\t1\tTrue\n",
      "9398\t9397\t1\tFalse\n",
      "9404\t9403\t1\tFalse\n",
      "9410\t97\t2\tTrue\n",
      "9414\t9413\t1\tFalse\n",
      "9420\t9419\t1\tTrue\n",
      "9422\t9421\t1\tTrue\n",
      "9432\t9431\t1\tTrue\n",
      "9434\t9433\t1\tFalse\n",
      "9438\t9437\t1\tFalse\n",
      "9440\t9439\t1\tTrue\n",
      "9462\t9461\t1\tTrue\n",
      "9464\t9463\t1\tFalse\n",
      "9468\t9467\t1\tFalse\n",
      "9474\t9473\t1\tFalse\n",
      "9480\t9479\t1\tTrue\n",
      "9492\t9491\t1\tTrue\n",
      "9498\t9497\t1\tFalse\n",
      "9512\t9511\t1\tTrue\n",
      "9522\t9521\t1\tTrue\n",
      "9534\t9533\t1\tFalse\n",
      "9540\t9539\t1\tTrue\n",
      "9548\t9547\t1\tFalse\n",
      "9552\t9551\t1\tTrue\n",
      "9588\t9587\t1\tFalse\n",
      "9602\t9601\t1\tTrue\n",
      "9614\t9613\t1\tFalse\n",
      "9620\t9619\t1\tTrue\n",
      "9624\t9623\t1\tFalse\n",
      "9630\t9629\t1\tTrue\n",
      "9632\t9631\t1\tTrue\n",
      "9644\t9643\t1\tFalse\n",
      "9650\t9649\t1\tTrue\n",
      "9662\t9661\t1\tTrue\n",
      "9678\t9677\t1\tFalse\n",
      "9680\t9679\t1\tTrue\n",
      "9690\t9689\t1\tTrue\n",
      "9698\t9697\t1\tFalse\n",
      "9720\t9719\t1\tTrue\n",
      "9722\t9721\t1\tTrue\n",
      "9734\t9733\t1\tFalse\n",
      "9740\t9739\t1\tTrue\n",
      "9744\t9743\t1\tFalse\n",
      "9750\t9749\t1\tTrue\n",
      "9768\t9767\t1\tFalse\n",
      "9770\t9769\t1\tTrue\n",
      "9782\t9781\t1\tTrue\n",
      "9788\t9787\t1\tFalse\n",
      "9792\t9791\t1\tTrue\n",
      "9804\t9803\t1\tFalse\n",
      "9812\t9811\t1\tTrue\n",
      "9818\t9817\t1\tFalse\n",
      "9830\t9829\t1\tTrue\n",
      "9834\t9833\t1\tFalse\n",
      "9840\t9839\t1\tTrue\n",
      "9852\t9851\t1\tTrue\n",
      "9858\t9857\t1\tFalse\n",
      "9860\t9859\t1\tTrue\n",
      "9872\t9871\t1\tTrue\n",
      "9884\t9883\t1\tFalse\n",
      "9888\t9887\t1\tFalse\n",
      "9902\t9901\t1\tTrue\n",
      "9908\t9907\t1\tFalse\n",
      "9924\t9923\t1\tFalse\n",
      "9930\t9929\t1\tTrue\n",
      "9932\t9931\t1\tTrue\n",
      "9942\t9941\t1\tTrue\n",
      "9950\t9949\t1\tTrue\n",
      "9968\t9967\t1\tFalse\n",
      "9974\t9973\t1\tFalse\n"
     ]
    }
   ],
   "source": [
    "#予想'の検証\n",
    "N=10000\n",
    "R = []\n",
    "for deg in range(0,N+1,2):\n",
    "    if(dim(deg//2)==0):\n",
    "        continue\n",
    "    if(is_prime_power(deg-1)):\n",
    "        p,k = factor(deg-1)[0]\n",
    "    else:\n",
    "        continue\n",
    "    Ord = [0 for i in range(deg+1)] #何回割れるか\n",
    "    for m in range(1,k+1):\n",
    "        f = awesome_polynomial_mod(deg, p^m)\n",
    "        #show(f)\n",
    "        for i, c in enumerate(f.list()):\n",
    "            if(c == 0):\n",
    "                Ord[i] = m;\n",
    "    #print(Ord)\n",
    "    ok = all([ord_p(i,p)+ord_p(deg-i,p)+r >= k for i,r in enumerate(Ord)])\n",
    "    if(ok):\n",
    "        R.append(deg-1)\n",
    "    print(f\"{deg}\\t{p}\\t{k}\\t{ok}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q = list(filter(lambda q: is_prime_power(q) and (q%10==1 or q%10==9), range(1, N+1, 2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{9, 81, 729, 6561}"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(Q)-set(R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "set()"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(R)-set(Q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{730} + X^{670} + X^{660} + X^{655} + X^{645} + 2 X^{640} + 2 X^{630} + 2 X^{625} + 2 X^{615} + X^{610} + X^{600} + X^{535} + X^{525} + X^{520} + X^{510} + 2 X^{505} + 2 X^{495} + 2 X^{490} + 2 X^{480} + X^{475} + X^{465} + 2 X^{400} + 2 X^{390} + 2 X^{385} + 2 X^{375} + X^{370} + X^{360} + X^{355} + X^{345} + 2 X^{340} + 2 X^{330} + 2 X^{265} + 2 X^{255} + 2 X^{250} + 2 X^{240} + X^{235} + X^{225} + X^{220} + X^{210} + 2 X^{205} + 2 X^{195} + X^{130} + X^{120} + X^{115} + X^{105} + 2 X^{100} + 2 X^{90} + 2 X^{85} + 2 X^{75} + X^{70} + X^{60} + 1"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(730, 3)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_prime(221)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_prime(211)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{211} + 210 X"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(212, 211)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{221} + 4 X^{196} + 2 X^{156} + 8 X^{131} + 5 X^{91} + 2 X^{66} + 4 X^{26} + 12 X"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(222, 13)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{221} + 9 X^{171} + 12 X^{136} + 12 X^{86} + 8 X^{51} + 16 X"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(222, 17)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{244} + 6 X^{224} + 9 X^{219} + 10 X^{214} + 5 X^{209} + 12 X^{204} + 12 X^{199} + X^{194} + 11 X^{189} + 7 X^{184} + 10 X^{179} + 10 X^{174} + 6 X^{169} + 11 X^{164} + 12 X^{159} + 7 X^{154} + 12 X^{149} + 6 X^{144} + 7 X^{129} + 12 X^{124} + 6 X^{119} + 6 X^{104} + X^{99} + 7 X^{94} + X^{89} + 11 X^{84} + 7 X^{79} + 10 X^{74} + 3 X^{69} + 7 X^{64} + 2 X^{59} + X^{54} + X^{49} + 12 X^{44} + 8 X^{39} + 10 X^{34} + 4 X^{29} + 6 X^{24} + X^{4}"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(248, 13)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{244} + 10 X^{224} + 12 X^{219} + 9 X^{214} + 14 X^{209} + 14 X^{204} + 12 X^{199} + 5 X^{194} + 17 X^{189} + 9 X^{184} + 14 X^{179} + 6 X^{174} + 2 X^{169} + 7 X^{164} + 17 X^{159} + 2 X^{154} + 17 X^{149} + 14 X^{144} + 9 X^{139} + 14 X^{134} + 12 X^{129} + 3 X^{124} + 7 X^{119} + 14 X^{114} + 10 X^{109} + 14 X^{104} + 2 X^{99} + 2 X^{94} + 2 X^{89} + 7 X^{84} + 17 X^{79} + 6 X^{74} + 5 X^{69} + 9 X^{64} + 2 X^{59} + 5 X^{54} + 7 X^{49} + 14 X^{44} + 5 X^{39} + 9 X^{34} + 7 X^{29} + 10 X^{24} + X^{4}"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(248, 19)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{1001} + 6 X^{911} + 3 X^{876} + 6 X^{861} + 6 X^{841} + 3 X^{826} + 3 X^{806} + 6 X^{791} + 5 X^{771} + 3 X^{756} + 2 X^{736} + 5 X^{721} + 6 X^{701} + 2 X^{686} + 2 X^{651} + 4 X^{631} + 4 X^{596} + 4 X^{581} + 6 X^{561} + 4 X^{546} + 6 X^{526} + 6 X^{511} + X^{491} + 6 X^{476} + 4 X^{456} + X^{441} + 3 X^{421} + 4 X^{406} + 3 X^{371} + 5 X^{351} + 2 X^{316} + X^{301} + 2 X^{281} + 2 X^{266} + 3 X^{246} + 2 X^{231} + X^{211} + 3 X^{196} + 3 X^{176} + X^{161} + X^{141} + 3 X^{126} + X^{91} + 6 X"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(1002, 7)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{1001} + 3 X^{891} + 10 X^{881} + 3 X^{781} + 8 X^{771} + X^{726} + 3 X^{671} + 8 X^{661} + 9 X^{616} + 10 X^{606} + 3 X^{561} + 8 X^{551} + 2 X^{506} + 2 X^{496} + 3 X^{451} + 8 X^{441} + 10 X^{396} + 9 X^{386} + 3 X^{341} + 8 X^{331} + X^{276} + 3 X^{231} + 8 X^{221} + X^{121} + 8 X^{111} + 10 X"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(1002, 11)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^{1001} + 7 X^{911} + 10 X^{871} + 10 X^{846} + X^{781} + 4 X^{741} + 10 X^{716} + 2 X^{676} + 2 X^{651} + 11 X^{611} + 6 X^{546} + 2 X^{521} + 11 X^{481} + 6 X^{456} + 2 X^{391} + 11 X^{351} + 2 X^{326} + 10 X^{286} + 9 X^{261} + 12 X^{221} + 10 X^{156} + 3 X^{131} + 6 X^{91} + 12 X"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = awesome_polynomial_mod(1002, 13)\n",
    "latex(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [],
   "source": [
    "def show_latex_polynomial_mod(deg,m):\n",
    "    f = awesome_polynomial_mod(deg, m)\n",
    "    c = f.list()\n",
    "    #deg = f.degree()\n",
    "    i_r = deg\n",
    "    i = 0\n",
    "    while(i_r>=i):\n",
    "        if(i_r >= len(c)):\n",
    "            i_r-=1\n",
    "            i+=1\n",
    "            continue\n",
    "        if(c[i]==0 and c[i_r]==0):\n",
    "            i_r-=1\n",
    "            i+=1\n",
    "            continue\n",
    "        #print(c[i], c[i_r])\n",
    "        coef_str = \"\" if c[i_r]==1 else c[i_r]\n",
    "        if(i==i_r):\n",
    "            print(f\"+{coef_str}x^{{{i_r}}}y^{{{i}}}\")\n",
    "        if(Mod(c[i]-c[i_r],m)==0):\n",
    "            print(f\"+{coef_str}(x^{{{i_r}}}y^{{{i}}}+x^{{{i}}}y^{{{i_r}}})\")\n",
    "        elif(Mod(c[i]+c[i_r],m)==0):\n",
    "            print(f\"+{coef_str}(x^{{{i_r}}}y^{{{i}}}-x^{{{i}}}y^{{{i_r}}})\")\n",
    "        i_r-=1\n",
    "        i+=1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+(x^{1001}y^{1}-x^{1}y^{1001})\n",
      "+7(x^{911}y^{91}-x^{91}y^{911})\n",
      "+10(x^{871}y^{131}-x^{131}y^{871})\n",
      "+10(x^{846}y^{156}+x^{156}y^{846})\n",
      "+(x^{781}y^{221}-x^{221}y^{781})\n",
      "+4(x^{741}y^{261}-x^{261}y^{741})\n",
      "+10(x^{716}y^{286}+x^{286}y^{716})\n",
      "+2(x^{676}y^{326}+x^{326}y^{676})\n",
      "+2(x^{651}y^{351}-x^{351}y^{651})\n",
      "+11(x^{611}y^{391}-x^{391}y^{611})\n",
      "+6(x^{546}y^{456}+x^{456}y^{546})\n",
      "+2(x^{521}y^{481}-x^{481}y^{521})\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial_mod(1002,13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+(x^{1001}y^{1}-x^{1}y^{1001})\n",
      "+6(x^{911}y^{91}-x^{91}y^{911})\n",
      "+3(x^{876}y^{126}+x^{126}y^{876})\n",
      "+6(x^{861}y^{141}-x^{141}y^{861})\n",
      "+6(x^{841}y^{161}-x^{161}y^{841})\n",
      "+3(x^{826}y^{176}+x^{176}y^{826})\n",
      "+3(x^{806}y^{196}+x^{196}y^{806})\n",
      "+6(x^{791}y^{211}-x^{211}y^{791})\n",
      "+5(x^{771}y^{231}-x^{231}y^{771})\n",
      "+3(x^{756}y^{246}+x^{246}y^{756})\n",
      "+2(x^{736}y^{266}+x^{266}y^{736})\n",
      "+5(x^{721}y^{281}-x^{281}y^{721})\n",
      "+6(x^{701}y^{301}-x^{301}y^{701})\n",
      "+2(x^{686}y^{316}+x^{316}y^{686})\n",
      "+2(x^{651}y^{351}-x^{351}y^{651})\n",
      "+4(x^{631}y^{371}-x^{371}y^{631})\n",
      "+4(x^{596}y^{406}+x^{406}y^{596})\n",
      "+4(x^{581}y^{421}-x^{421}y^{581})\n",
      "+6(x^{561}y^{441}-x^{441}y^{561})\n",
      "+4(x^{546}y^{456}+x^{456}y^{546})\n",
      "+6(x^{526}y^{476}+x^{476}y^{526})\n",
      "+6(x^{511}y^{491}-x^{491}y^{511})\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial_mod(1002,7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+(x^{1001}y^{1}-x^{1}y^{1001})\n",
      "+3(x^{891}y^{111}-x^{111}y^{891})\n",
      "+10(x^{881}y^{121}-x^{121}y^{881})\n",
      "+3(x^{781}y^{221}-x^{221}y^{781})\n",
      "+8(x^{771}y^{231}-x^{231}y^{771})\n",
      "+(x^{726}y^{276}+x^{276}y^{726})\n",
      "+3(x^{671}y^{331}-x^{331}y^{671})\n",
      "+8(x^{661}y^{341}-x^{341}y^{661})\n",
      "+9(x^{616}y^{386}+x^{386}y^{616})\n",
      "+10(x^{606}y^{396}+x^{396}y^{606})\n",
      "+3(x^{561}y^{441}-x^{441}y^{561})\n",
      "+8(x^{551}y^{451}-x^{451}y^{551})\n",
      "+2(x^{506}y^{496}+x^{496}y^{506})\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial_mod(1002,11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+(x^{211}y^{1}-x^{1}y^{211})\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial_mod(212,211)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+(x^{221}y^{1}-x^{1}y^{221})\n",
      "+17(x^{196}y^{26}+x^{26}y^{196})\n",
      "+26(x^{171}y^{51}-x^{51}y^{171})\n",
      "+119(x^{156}y^{66}+x^{66}y^{156})\n",
      "+182(x^{136}y^{86}+x^{86}y^{136})\n",
      "+34(x^{131}y^{91}-x^{91}y^{131})\n"
     ]
    }
   ],
   "source": [
    "show_latex_polynomial_mod(222,221)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle +\\left(X^{221} - X\\right)\n",
       "-13 \\!\\cdot\\! 17 \\!\\cdot\\! 23 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 269\\left(X^{201} - X^{21}\\right)\n",
       "+3^3 \\!\\cdot\\! 11 \\!\\cdot\\! 17 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 307 \\!\\cdot\\! 2591\\left(X^{196} + X^{26}\\right)\n",
       "-2^4 \\!\\cdot\\! 5^2 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 73 \\!\\cdot\\! 3593 \\!\\cdot\\! 410093\\left(X^{191} - X^{31}\\right)\n",
       "+2 \\!\\cdot\\! 5^3 \\!\\cdot\\! 11 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 19 \\!\\cdot\\! 73 \\!\\cdot\\! 4527644353\\left(X^{186} + X^{36}\\right)\n",
       "-3^2 \\!\\cdot\\! 5^2 \\!\\cdot\\! 7 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 23 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 35761534091\\left(X^{181} - X^{41}\\right)\n",
       "+3^2 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 31 \\!\\cdot\\! 71 \\!\\cdot\\! 73 \\!\\cdot\\! 89 \\!\\cdot\\! 266897 \\!\\cdot\\! 6485081\\left(X^{176} + X^{46}\\right)\n",
       "-2 \\!\\cdot\\! 13 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 28786509283165024681\\left(X^{171} - X^{51}\\right)\n",
       "+5^2 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 19 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 191 \\!\\cdot\\! 397 \\!\\cdot\\! 4091 \\!\\cdot\\! 322400791\\left(X^{166} + X^{56}\\right)\n",
       "-3^2 \\!\\cdot\\! 5^3 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 31 \\!\\cdot\\! 73^2 \\!\\cdot\\! 448421 \\!\\cdot\\! 1295630491\\left(X^{161} - X^{61}\\right)\n",
       "-3 \\!\\cdot\\! 5^2 \\!\\cdot\\! 17 \\!\\cdot\\! 23 \\!\\cdot\\! 73 \\!\\cdot\\! 113 \\!\\cdot\\! 1423 \\!\\cdot\\! 809399 \\!\\cdot\\! 9450302369\\left(X^{156} + X^{66}\\right)\n",
       "+2^2 \\!\\cdot\\! 3^2 \\!\\cdot\\! 7 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 19 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 33018649 \\!\\cdot\\! 345758436403\\left(X^{151} - X^{71}\\right)\n",
       "-3^2 \\!\\cdot\\! 7 \\!\\cdot\\! 11 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 19 \\!\\cdot\\! 31 \\!\\cdot\\! 151 \\!\\cdot\\! 643 \\!\\cdot\\! 14155299206458343\\left(X^{146} + X^{76}\\right)\n",
       "+2 \\!\\cdot\\! 5^3 \\!\\cdot\\! 13^3 \\!\\cdot\\! 17 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 151 \\!\\cdot\\! 94634324943376649\\left(X^{141} - X^{81}\\right)\n",
       "-5^4 \\!\\cdot\\! 11 \\!\\cdot\\! 13 \\!\\cdot\\! 23 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 151 \\!\\cdot\\! 5407 \\!\\cdot\\! 111631024687993\\left(X^{136} + X^{86}\\right)\n",
       "+2 \\!\\cdot\\! 3 \\!\\cdot\\! 5^3 \\!\\cdot\\! 17 \\!\\cdot\\! 19 \\!\\cdot\\! 23 \\!\\cdot\\! 31^2 \\!\\cdot\\! 73 \\!\\cdot\\! 151 \\!\\cdot\\! 62270573 \\!\\cdot\\! 96855581\\left(X^{131} - X^{91}\\right)\n",
       "-2^3 \\!\\cdot\\! 11 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 19 \\!\\cdot\\! 73 \\!\\cdot\\! 131 \\!\\cdot\\! 151 \\!\\cdot\\! 13663501 \\!\\cdot\\! 25039673107\\left(X^{126} + X^{96}\\right)\n",
       "+2 \\!\\cdot\\! 3 \\!\\cdot\\! 5 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 31 \\!\\cdot\\! 71 \\!\\cdot\\! 73 \\!\\cdot\\! 131 \\!\\cdot\\! 151 \\!\\cdot\\! 2761281432552953\\left(X^{121} - X^{101}\\right)\n",
       "-2 \\!\\cdot\\! 3^3 \\!\\cdot\\! 5^3 \\!\\cdot\\! 11 \\!\\cdot\\! 13 \\!\\cdot\\! 17 \\!\\cdot\\! 31 \\!\\cdot\\! 73 \\!\\cdot\\! 131 \\!\\cdot\\! 151 \\!\\cdot\\! 197 \\!\\cdot\\! 5807 \\!\\cdot\\! 13874537\\left(X^{116} + X^{106}\\right)\n",
       "$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = awesome_polynomial(222)\n",
    "show_latex_polynomial(f,222,xy=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "12\t11\tTrue\n",
      "20\t19\tTrue\n",
      "24\t23\tFalse\n",
      "30\t29\tTrue\n",
      "32\t31\tTrue\n",
      "36\t5 * 7\tFalse\n",
      "40\t3 * 13\tFalse\n",
      "42\t41\tTrue\n",
      "44\t43\tFalse\n",
      "48\t47\tFalse\n",
      "50\t7^2\tTrue\n",
      "52\t3 * 17\tFalse\n",
      "54\t53\tFalse\n",
      "56\t5 * 11\tFalse\n",
      "60\t59\tTrue\n",
      "62\t61\tTrue\n",
      "64\t3^2 * 7\tFalse\n",
      "66\t5 * 13\tFalse\n",
      "68\t67\tFalse\n",
      "70\t3 * 23\tFalse\n",
      "72\t71\tTrue\n",
      "74\t73\tFalse\n",
      "76\t3 * 5^2\tFalse\n",
      "78\t7 * 11\tFalse\n",
      "80\t79\tTrue\n",
      "82\t3^4\tFalse\n",
      "84\t83\tFalse\n",
      "86\t5 * 17\tFalse\n",
      "88\t3 * 29\tFalse\n",
      "90\t89\tTrue\n",
      "92\t7 * 13\tTrue\n",
      "94\t3 * 31\tFalse\n",
      "96\t5 * 19\tFalse\n",
      "98\t97\tFalse\n",
      "100\t3^2 * 11\tFalse\n",
      "102\t101\tTrue\n",
      "104\t103\tFalse\n",
      "106\t3 * 5 * 7\tFalse\n",
      "108\t107\tFalse\n",
      "110\t109\tTrue\n",
      "112\t3 * 37\tFalse\n",
      "114\t113\tFalse\n",
      "116\t5 * 23\tFalse\n",
      "118\t3^2 * 13\tFalse\n",
      "120\t7 * 17\tTrue\n",
      "122\t11^2\tTrue\n",
      "124\t3 * 41\tFalse\n",
      "126\t5^3\tFalse\n",
      "128\t127\tFalse\n",
      "130\t3 * 43\tFalse\n",
      "132\t131\tTrue\n",
      "134\t7 * 19\tFalse\n",
      "136\t3^3 * 5\tFalse\n",
      "138\t137\tFalse\n",
      "140\t139\tTrue\n",
      "142\t3 * 47\tFalse\n",
      "144\t11 * 13\tFalse\n",
      "146\t5 * 29\tFalse\n",
      "148\t3 * 7^2\tFalse\n",
      "150\t149\tTrue\n",
      "152\t151\tTrue\n",
      "154\t3^2 * 17\tFalse\n",
      "156\t5 * 31\tFalse\n",
      "158\t157\tFalse\n",
      "160\t3 * 53\tFalse\n",
      "162\t7 * 23\tTrue\n",
      "164\t163\tFalse\n",
      "166\t3 * 5 * 11\tFalse\n",
      "168\t167\tFalse\n",
      "170\t13^2\tTrue\n",
      "172\t3^2 * 19\tFalse\n",
      "174\t173\tFalse\n",
      "176\t5^2 * 7\tFalse\n",
      "178\t3 * 59\tFalse\n",
      "180\t179\tTrue\n",
      "182\t181\tTrue\n",
      "184\t3 * 61\tFalse\n",
      "186\t5 * 37\tFalse\n",
      "188\t11 * 17\tFalse\n",
      "190\t3^3 * 7\tFalse\n",
      "192\t191\tTrue\n",
      "194\t193\tFalse\n",
      "196\t3 * 5 * 13\tFalse\n",
      "198\t197\tFalse\n",
      "200\t199\tTrue\n",
      "202\t3 * 67\tFalse\n",
      "204\t7 * 29\tFalse\n",
      "206\t5 * 41\tFalse\n",
      "208\t3^2 * 23\tFalse\n",
      "210\t11 * 19\tTrue\n",
      "212\t211\tTrue\n",
      "214\t3 * 71\tFalse\n",
      "216\t5 * 43\tFalse\n",
      "218\t7 * 31\tFalse\n",
      "220\t3 * 73\tFalse\n",
      "222\t13 * 17\tTrue\n",
      "224\t223\tFalse\n",
      "226\t3^2 * 5^2\tFalse\n",
      "228\t227\tFalse\n",
      "230\t229\tTrue\n",
      "232\t3 * 7 * 11\tFalse\n",
      "234\t233\tFalse\n",
      "236\t5 * 47\tFalse\n",
      "238\t3 * 79\tFalse\n",
      "240\t239\tTrue\n",
      "242\t241\tTrue\n",
      "244\t3^5\tFalse\n",
      "246\t5 * 7^2\tFalse\n",
      "248\t13 * 19\tFalse\n",
      "250\t3 * 83\tFalse\n",
      "252\t251\tTrue\n",
      "254\t11 * 23\tFalse\n",
      "256\t3 * 5 * 17\tFalse\n",
      "258\t257\tFalse\n",
      "260\t7 * 37\tTrue\n",
      "262\t3^2 * 29\tFalse\n",
      "264\t263\tFalse\n",
      "266\t5 * 53\tFalse\n",
      "268\t3 * 89\tFalse\n",
      "270\t269\tTrue\n",
      "272\t271\tTrue\n",
      "274\t3 * 7 * 13\tFalse\n",
      "276\t5^2 * 11\tFalse\n",
      "278\t277\tFalse\n",
      "280\t3^2 * 31\tFalse\n",
      "282\t281\tTrue\n",
      "284\t283\tFalse\n",
      "286\t3 * 5 * 19\tFalse\n",
      "288\t7 * 41\tFalse\n",
      "290\t17^2\tTrue\n",
      "292\t3 * 97\tFalse\n",
      "294\t293\tFalse\n",
      "296\t5 * 59\tFalse\n",
      "298\t3^3 * 11\tFalse\n",
      "300\t13 * 23\tTrue\n",
      "302\t7 * 43\tTrue\n",
      "304\t3 * 101\tFalse\n",
      "306\t5 * 61\tFalse\n",
      "308\t307\tFalse\n",
      "310\t3 * 103\tFalse\n",
      "312\t311\tTrue\n",
      "314\t313\tFalse\n",
      "316\t3^2 * 5 * 7\tFalse\n",
      "318\t317\tFalse\n",
      "320\t11 * 29\tTrue\n",
      "322\t3 * 107\tFalse\n",
      "324\t17 * 19\tFalse\n",
      "326\t5^2 * 13\tFalse\n",
      "328\t3 * 109\tFalse\n",
      "330\t7 * 47\tTrue\n",
      "332\t331\tTrue\n",
      "334\t3^2 * 37\tFalse\n",
      "336\t5 * 67\tFalse\n",
      "338\t337\tFalse\n",
      "340\t3 * 113\tFalse\n",
      "342\t11 * 31\tTrue\n",
      "344\t7^3\tFalse\n",
      "346\t3 * 5 * 23\tFalse\n",
      "348\t347\tFalse\n",
      "350\t349\tTrue\n",
      "352\t3^3 * 13\tFalse\n",
      "354\t353\tFalse\n",
      "356\t5 * 71\tFalse\n",
      "358\t3 * 7 * 17\tFalse\n",
      "360\t359\tTrue\n",
      "362\t19^2\tTrue\n",
      "364\t3 * 11^2\tFalse\n",
      "366\t5 * 73\tFalse\n",
      "368\t367\tFalse\n",
      "370\t3^2 * 41\tFalse\n",
      "372\t7 * 53\tTrue\n",
      "374\t373\tFalse\n",
      "376\t3 * 5^3\tFalse\n",
      "378\t13 * 29\tFalse\n",
      "380\t379\tTrue\n",
      "382\t3 * 127\tFalse\n",
      "384\t383\tFalse\n",
      "386\t5 * 7 * 11\tFalse\n",
      "388\t3^2 * 43\tFalse\n",
      "390\t389\tTrue\n",
      "392\t17 * 23\tTrue\n",
      "394\t3 * 131\tFalse\n",
      "396\t5 * 79\tFalse\n",
      "398\t397\tFalse\n",
      "400\t3 * 7 * 19\tFalse\n",
      "402\t401\tTrue\n",
      "404\t13 * 31\tFalse\n",
      "406\t3^4 * 5\tFalse\n",
      "408\t11 * 37\tFalse\n",
      "410\t409\tTrue\n",
      "412\t3 * 137\tFalse\n",
      "414\t7 * 59\tFalse\n",
      "416\t5 * 83\tFalse\n",
      "418\t3 * 139\tFalse\n",
      "420\t419\tTrue\n",
      "422\t421\tTrue\n",
      "424\t3^2 * 47\tFalse\n",
      "426\t5^2 * 17\tFalse\n",
      "428\t7 * 61\tFalse\n",
      "430\t3 * 11 * 13\tTrue\n",
      "432\t431\tTrue\n",
      "434\t433\tFalse\n",
      "436\t3 * 5 * 29\tFalse\n",
      "438\t19 * 23\tFalse\n",
      "440\t439\tTrue\n",
      "442\t3^2 * 7^2\tFalse\n",
      "444\t443\tFalse\n",
      "446\t5 * 89\tFalse\n",
      "448\t3 * 149\tFalse\n",
      "450\t449\tTrue\n",
      "452\t11 * 41\tTrue\n",
      "454\t3 * 151\tFalse\n",
      "456\t5 * 7 * 13\tFalse\n",
      "458\t457\tFalse\n",
      "460\t3^3 * 17\tFalse\n",
      "462\t461\tTrue\n",
      "464\t463\tFalse\n",
      "466\t3 * 5 * 31\tFalse\n",
      "468\t467\tFalse\n",
      "470\t7 * 67\tTrue\n",
      "472\t3 * 157\tFalse\n",
      "474\t11 * 43\tFalse\n",
      "476\t5^2 * 19\tFalse\n",
      "478\t3^2 * 53\tFalse\n",
      "480\t479\tTrue\n",
      "482\t13 * 37\tTrue\n",
      "484\t3 * 7 * 23\tFalse\n",
      "486\t5 * 97\tFalse\n",
      "488\t487\tFalse\n",
      "490\t3 * 163\tFalse\n",
      "492\t491\tTrue\n",
      "494\t17 * 29\tFalse\n",
      "496\t3^2 * 5 * 11\tFalse\n",
      "498\t7 * 71\tFalse\n",
      "500\t499\tTrue\n",
      "502\t3 * 167\tFalse\n",
      "504\t503\tFalse\n",
      "506\t5 * 101\tFalse\n",
      "508\t3 * 13^2\tFalse\n",
      "510\t509\tTrue\n",
      "512\t7 * 73\tTrue\n",
      "514\t3^3 * 19\tFalse\n",
      "516\t5 * 103\tFalse\n",
      "518\t11 * 47\tFalse\n",
      "520\t3 * 173\tFalse\n",
      "522\t521\tTrue\n",
      "524\t523\tFalse\n",
      "526\t3 * 5^2 * 7\tFalse\n",
      "528\t17 * 31\tFalse\n",
      "530\t23^2\tTrue\n",
      "532\t3^2 * 59\tFalse\n",
      "534\t13 * 41\tFalse\n",
      "536\t5 * 107\tFalse\n",
      "538\t3 * 179\tFalse\n",
      "540\t7^2 * 11\tTrue\n",
      "542\t541\tTrue\n",
      "544\t3 * 181\tFalse\n",
      "546\t5 * 109\tFalse\n",
      "548\t547\tFalse\n",
      "550\t3^2 * 61\tFalse\n",
      "552\t19 * 29\tTrue\n",
      "554\t7 * 79\tFalse\n",
      "556\t3 * 5 * 37\tFalse\n",
      "558\t557\tFalse\n",
      "560\t13 * 43\tTrue\n",
      "562\t3 * 11 * 17\tTrue\n",
      "564\t563\tFalse\n",
      "566\t5 * 113\tFalse\n",
      "568\t3^4 * 7\tFalse\n",
      "570\t569\tTrue\n",
      "572\t571\tTrue\n",
      "574\t3 * 191\tFalse\n",
      "576\t5^2 * 23\tFalse\n",
      "578\t577\tFalse\n",
      "580\t3 * 193\tFalse\n",
      "582\t7 * 83\tTrue\n",
      "584\t11 * 53\tFalse\n",
      "586\t3^2 * 5 * 13\tFalse\n",
      "588\t587\tFalse\n",
      "590\t19 * 31\tTrue\n",
      "592\t3 * 197\tFalse\n",
      "594\t593\tFalse\n",
      "596\t5 * 7 * 17\tFalse\n",
      "598\t3 * 199\tFalse\n",
      "600\t599\tTrue\n",
      "602\t601\tTrue\n",
      "604\t3^2 * 67\tFalse\n",
      "606\t5 * 11^2\tFalse\n",
      "608\t607\tFalse\n",
      "610\t3 * 7 * 29\tFalse\n",
      "612\t13 * 47\tTrue\n",
      "614\t613\tFalse\n",
      "616\t3 * 5 * 41\tFalse\n",
      "618\t617\tFalse\n",
      "620\t619\tTrue\n",
      "622\t3^3 * 23\tFalse\n",
      "624\t7 * 89\tFalse\n",
      "626\t5^4\tFalse\n",
      "628\t3 * 11 * 19\tFalse\n",
      "630\t17 * 37\tTrue\n",
      "632\t631\tTrue\n",
      "634\t3 * 211\tFalse\n",
      "636\t5 * 127\tFalse\n",
      "638\t7^2 * 13\tFalse\n",
      "640\t3^2 * 71\tFalse\n",
      "642\t641\tTrue\n",
      "644\t643\tFalse\n",
      "646\t3 * 5 * 43\tFalse\n",
      "648\t647\tFalse\n",
      "650\t11 * 59\tTrue\n",
      "652\t3 * 7 * 31\tFalse\n",
      "654\t653\tFalse\n",
      "656\t5 * 131\tFalse\n",
      "658\t3^2 * 73\tFalse\n",
      "660\t659\tTrue\n",
      "662\t661\tTrue\n",
      "664\t3 * 13 * 17\tFalse\n",
      "666\t5 * 7 * 19\tFalse\n",
      "668\t23 * 29\tFalse\n",
      "670\t3 * 223\tFalse\n",
      "672\t11 * 61\tTrue\n",
      "674\t673\tFalse\n",
      "676\t3^3 * 5^2\tFalse\n",
      "678\t677\tFalse\n",
      "680\t7 * 97\tTrue\n",
      "682\t3 * 227\tFalse\n",
      "684\t683\tFalse\n",
      "686\t5 * 137\tFalse\n",
      "688\t3 * 229\tFalse\n",
      "690\t13 * 53\tTrue\n",
      "692\t691\tTrue\n",
      "694\t3^2 * 7 * 11\tFalse\n",
      "696\t5 * 139\tFalse\n",
      "698\t17 * 41\tFalse\n",
      "700\t3 * 233\tFalse\n",
      "702\t701\tTrue\n",
      "704\t19 * 37\tFalse\n",
      "706\t3 * 5 * 47\tFalse\n",
      "708\t7 * 101\tFalse\n",
      "710\t709\tTrue\n",
      "712\t3^2 * 79\tFalse\n",
      "714\t23 * 31\tFalse\n",
      "716\t5 * 11 * 13\tFalse\n",
      "718\t3 * 239\tFalse\n",
      "720\t719\tTrue\n",
      "722\t7 * 103\tTrue\n",
      "724\t3 * 241\tFalse\n",
      "726\t5^2 * 29\tFalse\n",
      "728\t727\tFalse\n",
      "730\t3^6\tFalse\n",
      "732\t17 * 43\tTrue\n",
      "734\t733\tFalse\n",
      "736\t3 * 5 * 7^2\tFalse\n",
      "738\t11 * 67\tFalse\n",
      "740\t739\tTrue\n",
      "742\t3 * 13 * 19\tTrue\n",
      "744\t743\tFalse\n",
      "746\t5 * 149\tFalse\n",
      "748\t3^2 * 83\tFalse\n",
      "750\t7 * 107\tTrue\n",
      "752\t751\tTrue\n",
      "754\t3 * 251\tFalse\n",
      "756\t5 * 151\tFalse\n",
      "758\t757\tFalse\n",
      "760\t3 * 11 * 23\tTrue\n",
      "762\t761\tTrue\n",
      "764\t7 * 109\tFalse\n",
      "766\t3^2 * 5 * 17\tFalse\n",
      "768\t13 * 59\tFalse\n",
      "770\t769\tTrue\n",
      "772\t3 * 257\tFalse\n",
      "774\t773\tFalse\n",
      "776\t5^2 * 31\tFalse\n",
      "778\t3 * 7 * 37\tFalse\n",
      "780\t19 * 41\tTrue\n",
      "782\t11 * 71\tTrue\n",
      "784\t3^3 * 29\tFalse\n",
      "786\t5 * 157\tFalse\n",
      "788\t787\tFalse\n",
      "790\t3 * 263\tFalse\n",
      "792\t7 * 113\tTrue\n",
      "794\t13 * 61\tFalse\n",
      "796\t3 * 5 * 53\tFalse\n",
      "798\t797\tFalse\n",
      "800\t17 * 47\tTrue\n",
      "802\t3^2 * 89\tFalse\n",
      "804\t11 * 73\tFalse\n",
      "806\t5 * 7 * 23\tFalse\n",
      "808\t3 * 269\tFalse\n",
      "810\t809\tTrue\n",
      "812\t811\tTrue\n",
      "814\t3 * 271\tFalse\n",
      "816\t5 * 163\tFalse\n",
      "818\t19 * 43\tFalse\n",
      "820\t3^2 * 7 * 13\tFalse\n",
      "822\t821\tTrue\n",
      "824\t823\tFalse\n",
      "826\t3 * 5^2 * 11\tFalse\n",
      "828\t827\tFalse\n",
      "830\t829\tTrue\n",
      "832\t3 * 277\tFalse\n",
      "834\t7^2 * 17\tFalse\n",
      "836\t5 * 167\tFalse\n",
      "838\t3^3 * 31\tFalse\n",
      "840\t839\tTrue\n",
      "842\t29^2\tTrue\n",
      "844\t3 * 281\tFalse\n",
      "846\t5 * 13^2\tFalse\n",
      "848\t7 * 11^2\tFalse\n",
      "850\t3 * 283\tFalse\n",
      "852\t23 * 37\tTrue\n",
      "854\t853\tFalse\n",
      "856\t3^2 * 5 * 19\tFalse\n",
      "858\t857\tFalse\n",
      "860\t859\tTrue\n",
      "862\t3 * 7 * 41\tFalse\n",
      "864\t863\tFalse\n",
      "866\t5 * 173\tFalse\n",
      "868\t3 * 17^2\tFalse\n",
      "870\t11 * 79\tTrue\n",
      "872\t13 * 67\tTrue\n",
      "874\t3^2 * 97\tFalse\n",
      "876\t5^3 * 7\tFalse\n",
      "878\t877\tFalse\n",
      "880\t3 * 293\tFalse\n",
      "882\t881\tTrue\n",
      "884\t883\tFalse\n",
      "886\t3 * 5 * 59\tFalse\n",
      "888\t887\tFalse\n",
      "890\t7 * 127\tTrue\n",
      "892\t3^4 * 11\tTrue\n",
      "894\t19 * 47\tFalse\n",
      "896\t5 * 179\tFalse\n",
      "898\t3 * 13 * 23\tFalse\n",
      "900\t29 * 31\tTrue\n",
      "902\t17 * 53\tTrue\n",
      "904\t3 * 7 * 43\tFalse\n",
      "906\t5 * 181\tFalse\n",
      "908\t907\tFalse\n",
      "910\t3^2 * 101\tFalse\n",
      "912\t911\tTrue\n",
      "914\t11 * 83\tFalse\n",
      "916\t3 * 5 * 61\tFalse\n",
      "918\t7 * 131\tFalse\n",
      "920\t919\tTrue\n",
      "922\t3 * 307\tFalse\n",
      "924\t13 * 71\tFalse\n",
      "926\t5^2 * 37\tFalse\n",
      "928\t3^2 * 103\tFalse\n",
      "930\t929\tTrue\n",
      "932\t7^2 * 19\tTrue\n",
      "934\t3 * 311\tFalse\n",
      "936\t5 * 11 * 17\tFalse\n",
      "938\t937\tFalse\n",
      "940\t3 * 313\tFalse\n",
      "942\t941\tTrue\n",
      "944\t23 * 41\tFalse\n",
      "946\t3^3 * 5 * 7\tFalse\n",
      "948\t947\tFalse\n",
      "950\t13 * 73\tTrue\n",
      "952\t3 * 317\tFalse\n",
      "954\t953\tFalse\n",
      "956\t5 * 191\tFalse\n",
      "958\t3 * 11 * 29\tFalse\n",
      "960\t7 * 137\tTrue\n",
      "962\t31^2\tTrue\n",
      "964\t3^2 * 107\tFalse\n",
      "966\t5 * 193\tFalse\n",
      "968\t967\tFalse\n",
      "970\t3 * 17 * 19\tTrue\n",
      "972\t971\tTrue\n",
      "974\t7 * 139\tFalse\n",
      "976\t3 * 5^2 * 13\tFalse\n",
      "978\t977\tFalse\n",
      "980\t11 * 89\tTrue\n",
      "982\t3^2 * 109\tFalse\n",
      "984\t983\tFalse\n",
      "986\t5 * 197\tFalse\n",
      "988\t3 * 7 * 47\tFalse\n",
      "990\t23 * 43\tTrue\n",
      "992\t991\tTrue\n",
      "994\t3 * 331\tFalse\n",
      "996\t5 * 199\tFalse\n",
      "998\t997\tFalse\n",
      "1000\t3^3 * 37\tFalse\n",
      "1002\t7 * 11 * 13\tTrue\n",
      "1004\t17 * 59\tFalse\n",
      "1006\t3 * 5 * 67\tFalse\n",
      "1008\t19 * 53\tFalse\n",
      "1010\t1009\tTrue\n",
      "1012\t3 * 337\tFalse\n",
      "1014\t1013\tFalse\n",
      "1016\t5 * 7 * 29\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1018\t3^2 * 113\tFalse\n",
      "1020\t1019\tTrue\n",
      "1022\t1021\tTrue\n",
      "1024\t3 * 11 * 31\tFalse\n",
      "1026\t5^2 * 41\tFalse\n",
      "1028\t13 * 79\tFalse\n",
      "1030\t3 * 7^3\tFalse\n",
      "1032\t1031\tTrue\n",
      "1034\t1033\tFalse\n",
      "1036\t3^2 * 5 * 23\tFalse\n",
      "1038\t17 * 61\tFalse\n",
      "1040\t1039\tTrue\n",
      "1042\t3 * 347\tFalse\n",
      "1044\t7 * 149\tFalse\n",
      "1046\t5 * 11 * 19\tFalse\n",
      "1048\t3 * 349\tFalse\n",
      "1050\t1049\tTrue\n",
      "1052\t1051\tTrue\n",
      "1054\t3^4 * 13\tFalse\n",
      "1056\t5 * 211\tFalse\n",
      "1058\t7 * 151\tFalse\n",
      "1060\t3 * 353\tFalse\n",
      "1062\t1061\tTrue\n",
      "1064\t1063\tFalse\n",
      "1066\t3 * 5 * 71\tFalse\n",
      "1068\t11 * 97\tFalse\n",
      "1070\t1069\tTrue\n",
      "1072\t3^2 * 7 * 17\tFalse\n",
      "1074\t29 * 37\tFalse\n",
      "1076\t5^2 * 43\tFalse\n",
      "1078\t3 * 359\tFalse\n",
      "1080\t13 * 83\tTrue\n",
      "1082\t23 * 47\tTrue\n",
      "1084\t3 * 19^2\tFalse\n",
      "1086\t5 * 7 * 31\tFalse\n",
      "1088\t1087\tFalse\n",
      "1090\t3^2 * 11^2\tFalse\n",
      "1092\t1091\tTrue\n",
      "1094\t1093\tFalse\n",
      "1096\t3 * 5 * 73\tFalse\n",
      "1098\t1097\tFalse\n",
      "1100\t7 * 157\tTrue\n",
      "1102\t3 * 367\tFalse\n",
      "1104\t1103\tFalse\n",
      "1106\t5 * 13 * 17\tFalse\n",
      "1108\t3^3 * 41\tFalse\n",
      "1110\t1109\tTrue\n",
      "1112\t11 * 101\tTrue\n",
      "1114\t3 * 7 * 53\tFalse\n",
      "1116\t5 * 223\tFalse\n",
      "1118\t1117\tFalse\n",
      "1120\t3 * 373\tFalse\n",
      "1122\t19 * 59\tTrue\n",
      "1124\t1123\tFalse\n",
      "1126\t3^2 * 5^3\tFalse\n",
      "1128\t7^2 * 23\tFalse\n",
      "1130\t1129\tTrue\n",
      "1132\t3 * 13 * 29\tFalse\n",
      "1134\t11 * 103\tFalse\n",
      "1136\t5 * 227\tFalse\n",
      "1138\t3 * 379\tFalse\n",
      "1140\t17 * 67\tTrue\n",
      "1142\t7 * 163\tTrue\n",
      "1144\t3^2 * 127\tFalse\n",
      "1146\t5 * 229\tFalse\n",
      "1148\t31 * 37\tFalse\n",
      "1150\t3 * 383\tFalse\n",
      "1152\t1151\tTrue\n",
      "1154\t1153\tFalse\n",
      "1156\t3 * 5 * 7 * 11\tFalse\n",
      "1158\t13 * 89\tFalse\n",
      "1160\t19 * 61\tTrue\n",
      "1162\t3^3 * 43\tFalse\n",
      "1164\t1163\tFalse\n",
      "1166\t5 * 233\tFalse\n",
      "1168\t3 * 389\tFalse\n",
      "1170\t7 * 167\tTrue\n",
      "1172\t1171\tTrue\n",
      "1174\t3 * 17 * 23\tFalse\n",
      "1176\t5^2 * 47\tFalse\n",
      "1178\t11 * 107\tFalse\n",
      "1180\t3^2 * 131\tFalse\n",
      "1182\t1181\tTrue\n",
      "1184\t7 * 13^2\tFalse\n",
      "1186\t3 * 5 * 79\tFalse\n",
      "1188\t1187\tFalse\n",
      "1190\t29 * 41\tTrue\n",
      "1192\t3 * 397\tFalse\n",
      "1194\t1193\tFalse\n",
      "1196\t5 * 239\tFalse\n",
      "1198\t3^2 * 7 * 19\tFalse\n",
      "1200\t11 * 109\tTrue\n",
      "1202\t1201\tTrue\n",
      "1204\t3 * 401\tFalse\n",
      "1206\t5 * 241\tFalse\n",
      "1208\t17 * 71\tFalse\n",
      "1210\t3 * 13 * 31\tTrue\n",
      "1212\t7 * 173\tTrue\n",
      "1214\t1213\tFalse\n",
      "1216\t3^5 * 5\tFalse\n",
      "1218\t1217\tFalse\n",
      "1220\t23 * 53\tTrue\n",
      "1222\t3 * 11 * 37\tTrue\n",
      "1224\t1223\tFalse\n",
      "1226\t5^2 * 7^2\tFalse\n",
      "1228\t3 * 409\tFalse\n",
      "1230\t1229\tTrue\n",
      "1232\t1231\tTrue\n",
      "1234\t3^2 * 137\tFalse\n",
      "1236\t5 * 13 * 19\tFalse\n",
      "1238\t1237\tFalse\n",
      "1240\t3 * 7 * 59\tFalse\n",
      "1242\t17 * 73\tTrue\n",
      "1244\t11 * 113\tFalse\n",
      "1246\t3 * 5 * 83\tFalse\n",
      "1248\t29 * 43\tFalse\n",
      "1250\t1249\tTrue\n",
      "1252\t3^2 * 139\tFalse\n",
      "1254\t7 * 179\tFalse\n",
      "1256\t5 * 251\tFalse\n",
      "1258\t3 * 419\tFalse\n",
      "1260\t1259\tTrue\n",
      "1262\t13 * 97\tTrue\n",
      "1264\t3 * 421\tFalse\n",
      "1266\t5 * 11 * 23\tFalse\n",
      "1268\t7 * 181\tFalse\n",
      "1270\t3^3 * 47\tFalse\n",
      "1272\t31 * 41\tTrue\n",
      "1274\t19 * 67\tFalse\n",
      "1276\t3 * 5^2 * 17\tFalse\n",
      "1278\t1277\tFalse\n",
      "1280\t1279\tTrue\n",
      "1282\t3 * 7 * 61\tFalse\n",
      "1284\t1283\tFalse\n",
      "1286\t5 * 257\tFalse\n",
      "1288\t3^2 * 11 * 13\tFalse\n",
      "1290\t1289\tTrue\n",
      "1292\t1291\tTrue\n",
      "1294\t3 * 431\tFalse\n",
      "1296\t5 * 7 * 37\tFalse\n",
      "1298\t1297\tFalse\n",
      "1300\t3 * 433\tFalse\n",
      "1302\t1301\tTrue\n",
      "1304\t1303\tFalse\n",
      "1306\t3^2 * 5 * 29\tFalse\n",
      "1308\t1307\tFalse\n",
      "1310\t7 * 11 * 17\tTrue\n",
      "1312\t3 * 19 * 23\tFalse\n",
      "1314\t13 * 101\tFalse\n",
      "1316\t5 * 263\tFalse\n",
      "1318\t3 * 439\tFalse\n",
      "1320\t1319\tTrue\n",
      "1322\t1321\tTrue\n",
      "1324\t3^3 * 7^2\tFalse\n",
      "1326\t5^2 * 53\tFalse\n",
      "1328\t1327\tFalse\n",
      "1330\t3 * 443\tFalse\n",
      "1332\t11^3\tTrue\n",
      "1334\t31 * 43\tFalse\n",
      "1336\t3 * 5 * 89\tFalse\n",
      "1338\t7 * 191\tFalse\n",
      "1340\t13 * 103\tTrue\n",
      "1342\t3^2 * 149\tFalse\n",
      "1344\t17 * 79\tFalse\n",
      "1346\t5 * 269\tFalse\n",
      "1348\t3 * 449\tFalse\n",
      "1350\t19 * 71\tTrue\n",
      "1352\t7 * 193\tTrue\n",
      "1354\t3 * 11 * 41\tFalse\n",
      "1356\t5 * 271\tFalse\n",
      "1358\t23 * 59\tFalse\n",
      "1360\t3^2 * 151\tFalse\n",
      "1362\t1361\tTrue\n",
      "1364\t29 * 47\tFalse\n",
      "1366\t3 * 5 * 7 * 13\tFalse\n",
      "1368\t1367\tFalse\n",
      "1370\t37^2\tTrue\n",
      "1372\t3 * 457\tFalse\n",
      "1374\t1373\tFalse\n",
      "1376\t5^3 * 11\tFalse\n",
      "1378\t3^4 * 17\tFalse\n",
      "1380\t7 * 197\tTrue\n",
      "1382\t1381\tTrue\n",
      "1384\t3 * 461\tFalse\n",
      "1386\t5 * 277\tFalse\n",
      "1388\t19 * 73\tFalse\n",
      "1390\t3 * 463\tFalse\n",
      "1392\t13 * 107\tTrue\n",
      "1394\t7 * 199\tFalse\n",
      "1396\t3^2 * 5 * 31\tFalse\n",
      "1398\t11 * 127\tFalse\n",
      "1400\t1399\tTrue\n",
      "1402\t3 * 467\tFalse\n",
      "1404\t23 * 61\tFalse\n",
      "1406\t5 * 281\tFalse\n",
      "1408\t3 * 7 * 67\tFalse\n",
      "1410\t1409\tTrue\n",
      "1412\t17 * 83\tTrue\n",
      "1414\t3^2 * 157\tFalse\n",
      "1416\t5 * 283\tFalse\n",
      "1418\t13 * 109\tFalse\n",
      "1420\t3 * 11 * 43\tTrue\n",
      "1422\t7^2 * 29\tTrue\n",
      "1424\t1423\tFalse\n",
      "1426\t3 * 5^2 * 19\tFalse\n",
      "1428\t1427\tFalse\n",
      "1430\t1429\tTrue\n",
      "1432\t3^3 * 53\tFalse\n",
      "1434\t1433\tFalse\n",
      "1436\t5 * 7 * 41\tFalse\n",
      "1438\t3 * 479\tFalse\n",
      "1440\t1439\tTrue\n",
      "1442\t11 * 131\tTrue\n",
      "1444\t3 * 13 * 37\tFalse\n",
      "1446\t5 * 17^2\tFalse\n",
      "1448\t1447\tFalse\n",
      "1450\t3^2 * 7 * 23\tFalse\n",
      "1452\t1451\tTrue\n",
      "1454\t1453\tFalse\n",
      "1456\t3 * 5 * 97\tFalse\n",
      "1458\t31 * 47\tFalse\n",
      "1460\t1459\tTrue\n",
      "1462\t3 * 487\tFalse\n",
      "1464\t7 * 11 * 19\tFalse\n",
      "1466\t5 * 293\tFalse\n",
      "1468\t3^2 * 163\tFalse\n",
      "1470\t13 * 113\tTrue\n",
      "1472\t1471\tTrue\n",
      "1474\t3 * 491\tFalse\n",
      "1476\t5^2 * 59\tFalse\n",
      "1478\t7 * 211\tFalse\n",
      "1480\t3 * 17 * 29\tFalse\n",
      "1482\t1481\tTrue\n",
      "1484\t1483\tFalse\n",
      "1486\t3^3 * 5 * 11\tFalse\n",
      "1488\t1487\tFalse\n",
      "1490\t1489\tTrue\n",
      "1492\t3 * 7 * 71\tFalse\n",
      "1494\t1493\tFalse\n",
      "1496\t5 * 13 * 23\tFalse\n",
      "1498\t3 * 499\tFalse\n",
      "1500\t1499\tTrue\n",
      "1502\t19 * 79\tTrue\n",
      "1504\t3^2 * 167\tFalse\n",
      "1506\t5 * 7 * 43\tFalse\n",
      "1508\t11 * 137\tFalse\n",
      "1510\t3 * 503\tFalse\n",
      "1512\t1511\tTrue\n",
      "1514\t17 * 89\tFalse\n",
      "1516\t3 * 5 * 101\tFalse\n",
      "1518\t37 * 41\tFalse\n",
      "1520\t7^2 * 31\tTrue\n",
      "1522\t3^2 * 13^2\tFalse\n",
      "1524\t1523\tFalse\n",
      "1526\t5^2 * 61\tFalse\n",
      "1528\t3 * 509\tFalse\n",
      "1530\t11 * 139\tTrue\n",
      "1532\t1531\tTrue\n",
      "1534\t3 * 7 * 73\tFalse\n",
      "1536\t5 * 307\tFalse\n",
      "1538\t29 * 53\tFalse\n",
      "1540\t3^4 * 19\tFalse\n",
      "1542\t23 * 67\tTrue\n",
      "1544\t1543\tFalse\n",
      "1546\t3 * 5 * 103\tFalse\n",
      "1548\t7 * 13 * 17\tFalse\n",
      "1550\t1549\tTrue\n",
      "1552\t3 * 11 * 47\tFalse\n",
      "1554\t1553\tFalse\n",
      "1556\t5 * 311\tFalse\n",
      "1558\t3^2 * 173\tFalse\n",
      "1560\t1559\tTrue\n",
      "1562\t7 * 223\tTrue\n",
      "1564\t3 * 521\tFalse\n",
      "1566\t5 * 313\tFalse\n",
      "1568\t1567\tFalse\n",
      "1570\t3 * 523\tFalse\n",
      "1572\t1571\tTrue\n",
      "1574\t11^2 * 13\tFalse\n",
      "1576\t3^2 * 5^2 * 7\tFalse\n",
      "1578\t19 * 83\tFalse\n",
      "1580\t1579\tTrue\n",
      "1582\t3 * 17 * 31\tTrue\n",
      "1584\t1583\tFalse\n",
      "1586\t5 * 317\tFalse\n",
      "1588\t3 * 23^2\tFalse\n",
      "1590\t7 * 227\tTrue\n",
      "1592\t37 * 43\tTrue\n",
      "1594\t3^3 * 59\tFalse\n",
      "1596\t5 * 11 * 29\tFalse\n",
      "1598\t1597\tFalse\n",
      "1600\t3 * 13 * 41\tFalse\n",
      "1602\t1601\tTrue\n",
      "1604\t7 * 229\tFalse\n",
      "1606\t3 * 5 * 107\tFalse\n",
      "1608\t1607\tFalse\n",
      "1610\t1609\tTrue\n",
      "1612\t3^2 * 179\tFalse\n",
      "1614\t1613\tFalse\n",
      "1616\t5 * 17 * 19\tFalse\n",
      "1618\t3 * 7^2 * 11\tFalse\n",
      "1620\t1619\tTrue\n",
      "1622\t1621\tTrue\n",
      "1624\t3 * 541\tFalse\n",
      "1626\t5^3 * 13\tFalse\n",
      "1628\t1627\tFalse\n",
      "1630\t3^2 * 181\tFalse\n",
      "1632\t7 * 233\tTrue\n",
      "1634\t23 * 71\tFalse\n",
      "1636\t3 * 5 * 109\tFalse\n",
      "1638\t1637\tFalse\n",
      "1640\t11 * 149\tTrue\n",
      "1642\t3 * 547\tFalse\n",
      "1644\t31 * 53\tFalse\n",
      "1646\t5 * 7 * 47\tFalse\n",
      "1648\t3^3 * 61\tFalse\n",
      "1650\t17 * 97\tTrue\n",
      "1652\t13 * 127\tTrue\n",
      "1654\t3 * 19 * 29\tFalse\n",
      "1656\t5 * 331\tFalse\n",
      "1658\t1657\tFalse\n",
      "1660\t3 * 7 * 79\tFalse\n",
      "1662\t11 * 151\tTrue\n",
      "1664\t1663\tFalse\n",
      "1666\t3^2 * 5 * 37\tFalse\n",
      "1668\t1667\tFalse\n",
      "1670\t1669\tTrue\n",
      "1672\t3 * 557\tFalse\n",
      "1674\t7 * 239\tFalse\n",
      "1676\t5^2 * 67\tFalse\n",
      "1678\t3 * 13 * 43\tFalse\n",
      "1680\t23 * 73\tTrue\n",
      "1682\t41^2\tTrue\n",
      "1684\t3^2 * 11 * 17\tFalse\n",
      "1686\t5 * 337\tFalse\n",
      "1688\t7 * 241\tFalse\n",
      "1690\t3 * 563\tFalse\n",
      "1692\t19 * 89\tTrue\n",
      "1694\t1693\tFalse\n",
      "1696\t3 * 5 * 113\tFalse\n",
      "1698\t1697\tFalse\n",
      "1700\t1699\tTrue\n",
      "1702\t3^5 * 7\tFalse\n",
      "1704\t13 * 131\tFalse\n",
      "1706\t5 * 11 * 31\tFalse\n",
      "1708\t3 * 569\tFalse\n",
      "1710\t1709\tTrue\n",
      "1712\t29 * 59\tTrue\n",
      "1714\t3 * 571\tFalse\n",
      "1716\t5 * 7^3\tFalse\n",
      "1718\t17 * 101\tFalse\n",
      "1720\t3^2 * 191\tFalse\n",
      "1722\t1721\tTrue\n",
      "1724\t1723\tFalse\n",
      "1726\t3 * 5^2 * 23\tFalse\n",
      "1728\t11 * 157\tFalse\n",
      "1730\t7 * 13 * 19\tTrue\n",
      "1732\t3 * 577\tFalse\n",
      "1734\t1733\tFalse\n",
      "1736\t5 * 347\tFalse\n",
      "1738\t3^2 * 193\tFalse\n",
      "1740\t37 * 47\tTrue\n",
      "1742\t1741\tTrue\n",
      "1744\t3 * 7 * 83\tFalse\n",
      "1746\t5 * 349\tFalse\n",
      "1748\t1747\tFalse\n",
      "1750\t3 * 11 * 53\tTrue\n",
      "1752\t17 * 103\tTrue\n",
      "1754\t1753\tFalse\n",
      "1756\t3^3 * 5 * 13\tFalse\n",
      "1758\t7 * 251\tFalse\n",
      "1760\t1759\tTrue\n",
      "1762\t3 * 587\tFalse\n",
      "1764\t41 * 43\tFalse\n",
      "1766\t5 * 353\tFalse\n",
      "1768\t3 * 19 * 31\tFalse\n",
      "1770\t29 * 61\tTrue\n",
      "1772\t7 * 11 * 23\tTrue\n",
      "1774\t3^2 * 197\tFalse\n",
      "1776\t5^2 * 71\tFalse\n",
      "1778\t1777\tFalse\n",
      "1780\t3 * 593\tFalse\n",
      "1782\t13 * 137\tTrue\n",
      "1784\t1783\tFalse\n",
      "1786\t3 * 5 * 7 * 17\tFalse\n",
      "1788\t1787\tFalse\n",
      "1790\t1789\tTrue\n",
      "1792\t3^2 * 199\tFalse\n",
      "1794\t11 * 163\tFalse\n",
      "1796\t5 * 359\tFalse\n",
      "1798\t3 * 599\tFalse\n",
      "1800\t7 * 257\tTrue\n",
      "1802\t1801\tTrue\n",
      "1804\t3 * 601\tFalse\n",
      "1806\t5 * 19^2\tFalse\n",
      "1808\t13 * 139\tFalse\n",
      "1810\t3^3 * 67\tFalse\n",
      "1812\t1811\tTrue\n",
      "1814\t7^2 * 37\tFalse\n",
      "1816\t3 * 5 * 11^2\tFalse\n",
      "1818\t23 * 79\tFalse\n",
      "1820\t17 * 107\tTrue\n",
      "1822\t3 * 607\tFalse\n",
      "1824\t1823\tFalse\n",
      "1826\t5^2 * 73\tFalse\n",
      "1828\t3^2 * 7 * 29\tFalse\n",
      "1830\t31 * 59\tTrue\n",
      "1832\t1831\tTrue\n",
      "1834\t3 * 13 * 47\tFalse\n",
      "1836\t5 * 367\tFalse\n",
      "1838\t11 * 167\tFalse\n",
      "1840\t3 * 613\tFalse\n",
      "1842\t7 * 263\tTrue\n",
      "1844\t19 * 97\tFalse\n",
      "1846\t3^2 * 5 * 41\tFalse\n",
      "1848\t1847\tFalse\n",
      "1850\t43^2\tTrue\n",
      "1852\t3 * 617\tFalse\n",
      "1854\t17 * 109\tFalse\n",
      "1856\t5 * 7 * 53\tFalse\n",
      "1858\t3 * 619\tFalse\n",
      "1860\t11 * 13^2\tTrue\n",
      "1862\t1861\tTrue\n",
      "1864\t3^4 * 23\tFalse\n",
      "1866\t5 * 373\tFalse\n",
      "1868\t1867\tFalse\n",
      "1870\t3 * 7 * 89\tFalse\n",
      "1872\t1871\tTrue\n",
      "1874\t1873\tFalse\n",
      "1876\t3 * 5^4\tFalse\n",
      "1878\t1877\tFalse\n",
      "1880\t1879\tTrue\n",
      "1882\t3^2 * 11 * 19\tTrue\n",
      "1884\t7 * 269\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1886\t5 * 13 * 29\tFalse\n",
      "1888\t3 * 17 * 37\tFalse\n",
      "1890\t1889\tTrue\n",
      "1892\t31 * 61\tTrue\n",
      "1894\t3 * 631\tFalse\n",
      "1896\t5 * 379\tFalse\n",
      "1898\t7 * 271\tFalse\n",
      "1900\t3^2 * 211\tFalse\n",
      "1902\t1901\tTrue\n",
      "1904\t11 * 173\tFalse\n",
      "1906\t3 * 5 * 127\tFalse\n",
      "1908\t1907\tFalse\n",
      "1910\t23 * 83\tTrue\n",
      "1912\t3 * 7^2 * 13\tTrue\n",
      "1914\t1913\tFalse\n",
      "1916\t5 * 383\tFalse\n",
      "1918\t3^3 * 71\tFalse\n",
      "1920\t19 * 101\tTrue\n",
      "1922\t17 * 113\tTrue\n",
      "1924\t3 * 641\tFalse\n",
      "1926\t5^2 * 7 * 11\tFalse\n",
      "1928\t41 * 47\tFalse\n",
      "1930\t3 * 643\tFalse\n",
      "1932\t1931\tTrue\n",
      "1934\t1933\tFalse\n",
      "1936\t3^2 * 5 * 43\tFalse\n",
      "1938\t13 * 149\tFalse\n",
      "1940\t7 * 277\tTrue\n",
      "1942\t3 * 647\tFalse\n",
      "1944\t29 * 67\tFalse\n",
      "1946\t5 * 389\tFalse\n",
      "1948\t3 * 11 * 59\tFalse\n",
      "1950\t1949\tTrue\n",
      "1952\t1951\tTrue\n",
      "1954\t3^2 * 7 * 31\tFalse\n",
      "1956\t5 * 17 * 23\tFalse\n",
      "1958\t19 * 103\tFalse\n",
      "1960\t3 * 653\tFalse\n",
      "1962\t37 * 53\tTrue\n",
      "1964\t13 * 151\tFalse\n",
      "1966\t3 * 5 * 131\tFalse\n",
      "1968\t7 * 281\tFalse\n",
      "1970\t11 * 179\tTrue\n",
      "1972\t3^3 * 73\tFalse\n",
      "1974\t1973\tFalse\n",
      "1976\t5^2 * 79\tFalse\n",
      "1978\t3 * 659\tFalse\n",
      "1980\t1979\tTrue\n",
      "1982\t7 * 283\tTrue\n",
      "1984\t3 * 661\tFalse\n",
      "1986\t5 * 397\tFalse\n",
      "1988\t1987\tFalse\n",
      "1990\t3^2 * 13 * 17\tTrue\n",
      "1992\t11 * 181\tTrue\n",
      "1994\t1993\tFalse\n",
      "1996\t3 * 5 * 7 * 19\tFalse\n",
      "1998\t1997\tFalse\n",
      "2000\t1999\tTrue\n",
      "2002\t3 * 23 * 29\tFalse\n",
      "2004\t2003\tFalse\n",
      "2006\t5 * 401\tFalse\n",
      "2008\t3^2 * 223\tFalse\n",
      "2010\t7^2 * 41\tTrue\n",
      "2012\t2011\tTrue\n",
      "2014\t3 * 11 * 61\tFalse\n",
      "2016\t5 * 13 * 31\tFalse\n",
      "2018\t2017\tFalse\n",
      "2020\t3 * 673\tFalse\n",
      "2022\t43 * 47\tTrue\n",
      "2024\t7 * 17^2\tFalse\n",
      "2026\t3^4 * 5^2\tFalse\n",
      "2028\t2027\tFalse\n",
      "2030\t2029\tTrue\n",
      "2032\t3 * 677\tFalse\n",
      "2034\t19 * 107\tFalse\n",
      "2036\t5 * 11 * 37\tFalse\n",
      "2038\t3 * 7 * 97\tFalse\n",
      "2040\t2039\tTrue\n",
      "2042\t13 * 157\tTrue\n",
      "2044\t3^2 * 227\tFalse\n",
      "2046\t5 * 409\tFalse\n",
      "2048\t23 * 89\tFalse\n",
      "2050\t3 * 683\tFalse\n",
      "2052\t7 * 293\tTrue\n",
      "2054\t2053\tFalse\n",
      "2056\t3 * 5 * 137\tFalse\n",
      "2058\t11^2 * 17\tFalse\n",
      "2060\t29 * 71\tTrue\n",
      "2062\t3^2 * 229\tFalse\n",
      "2064\t2063\tFalse\n",
      "2066\t5 * 7 * 59\tFalse\n",
      "2068\t3 * 13 * 53\tFalse\n",
      "2070\t2069\tTrue\n",
      "2072\t19 * 109\tTrue\n",
      "2074\t3 * 691\tFalse\n",
      "2076\t5^2 * 83\tFalse\n",
      "2078\t31 * 67\tFalse\n",
      "2080\t3^3 * 7 * 11\tTrue\n",
      "2082\t2081\tTrue\n",
      "2084\t2083\tFalse\n",
      "2086\t3 * 5 * 139\tFalse\n",
      "2088\t2087\tFalse\n",
      "2090\t2089\tTrue\n",
      "2092\t3 * 17 * 41\tFalse\n",
      "2094\t7 * 13 * 23\tFalse\n",
      "2096\t5 * 419\tFalse\n",
      "2098\t3^2 * 233\tFalse\n",
      "2100\t2099\tTrue\n",
      "2102\t11 * 191\tTrue\n",
      "2104\t3 * 701\tFalse\n",
      "2106\t5 * 421\tFalse\n",
      "2108\t7^2 * 43\tFalse\n",
      "2110\t3 * 19 * 37\tTrue\n",
      "2112\t2111\tTrue\n",
      "2114\t2113\tFalse\n",
      "2116\t3^2 * 5 * 47\tFalse\n",
      "2118\t29 * 73\tFalse\n",
      "2120\t13 * 163\tTrue\n",
      "2122\t3 * 7 * 101\tFalse\n",
      "2124\t11 * 193\tFalse\n",
      "2126\t5^3 * 17\tFalse\n",
      "2128\t3 * 709\tFalse\n",
      "2130\t2129\tTrue\n",
      "2132\t2131\tTrue\n",
      "2134\t3^3 * 79\tFalse\n",
      "2136\t5 * 7 * 61\tFalse\n",
      "2138\t2137\tFalse\n",
      "2140\t3 * 23 * 31\tFalse\n",
      "2142\t2141\tTrue\n",
      "2144\t2143\tFalse\n",
      "2146\t3 * 5 * 11 * 13\tFalse\n",
      "2148\t19 * 113\tFalse\n",
      "2150\t7 * 307\tTrue\n",
      "2152\t3^2 * 239\tFalse\n",
      "2154\t2153\tFalse\n",
      "2156\t5 * 431\tFalse\n",
      "2158\t3 * 719\tFalse\n",
      "2160\t17 * 127\tTrue\n",
      "2162\t2161\tTrue\n",
      "2164\t3 * 7 * 103\tFalse\n",
      "2166\t5 * 433\tFalse\n",
      "2168\t11 * 197\tFalse\n",
      "2170\t3^2 * 241\tFalse\n",
      "2172\t13 * 167\tTrue\n",
      "2174\t41 * 53\tFalse\n",
      "2176\t3 * 5^2 * 29\tFalse\n",
      "2178\t7 * 311\tFalse\n",
      "2180\t2179\tTrue\n",
      "2182\t3 * 727\tFalse\n",
      "2184\t37 * 59\tFalse\n",
      "2186\t5 * 19 * 23\tFalse\n",
      "2188\t3^7\tFalse\n",
      "2190\t11 * 199\tTrue\n",
      "2192\t7 * 313\tTrue\n",
      "2194\t3 * 17 * 43\tFalse\n",
      "2196\t5 * 439\tFalse\n",
      "2198\t13^3\tFalse\n",
      "2200\t3 * 733\tFalse\n",
      "2202\t31 * 71\tTrue\n",
      "2204\t2203\tFalse\n",
      "2206\t3^2 * 5 * 7^2\tFalse\n",
      "2208\t2207\tFalse\n",
      "2210\t47^2\tTrue\n",
      "2212\t3 * 11 * 67\tTrue\n",
      "2214\t2213\tFalse\n",
      "2216\t5 * 443\tFalse\n",
      "2218\t3 * 739\tFalse\n",
      "2220\t7 * 317\tTrue\n",
      "2222\t2221\tTrue\n",
      "2224\t3^2 * 13 * 19\tFalse\n",
      "2226\t5^2 * 89\tFalse\n",
      "2228\t17 * 131\tFalse\n",
      "2230\t3 * 743\tFalse\n",
      "2232\t23 * 97\tTrue\n",
      "2234\t7 * 11 * 29\tFalse\n",
      "2236\t3 * 5 * 149\tFalse\n",
      "2238\t2237\tFalse\n",
      "2240\t2239\tTrue\n",
      "2242\t3^3 * 83\tFalse\n",
      "2244\t2243\tFalse\n",
      "2246\t5 * 449\tFalse\n",
      "2248\t3 * 7 * 107\tFalse\n",
      "2250\t13 * 173\tTrue\n",
      "2252\t2251\tTrue\n",
      "2254\t3 * 751\tFalse\n",
      "2256\t5 * 11 * 41\tFalse\n",
      "2258\t37 * 61\tFalse\n",
      "2260\t3^2 * 251\tFalse\n",
      "2262\t7 * 17 * 19\tTrue\n",
      "2264\t31 * 73\tFalse\n",
      "2266\t3 * 5 * 151\tFalse\n",
      "2268\t2267\tFalse\n",
      "2270\t2269\tTrue\n",
      "2272\t3 * 757\tFalse\n",
      "2274\t2273\tFalse\n",
      "2276\t5^2 * 7 * 13\tFalse\n",
      "2278\t3^2 * 11 * 23\tFalse\n",
      "2280\t43 * 53\tTrue\n",
      "2282\t2281\tTrue\n",
      "2284\t3 * 761\tFalse\n",
      "2286\t5 * 457\tFalse\n",
      "2288\t2287\tFalse\n",
      "2290\t3 * 7 * 109\tFalse\n",
      "2292\t29 * 79\tTrue\n",
      "2294\t2293\tFalse\n",
      "2296\t3^3 * 5 * 17\tFalse\n",
      "2298\t2297\tFalse\n",
      "2300\t11^2 * 19\tTrue\n",
      "2302\t3 * 13 * 59\tTrue\n",
      "2304\t7^2 * 47\tFalse\n",
      "2306\t5 * 461\tFalse\n",
      "2308\t3 * 769\tFalse\n",
      "2310\t2309\tTrue\n",
      "2312\t2311\tTrue\n",
      "2314\t3^2 * 257\tFalse\n",
      "2316\t5 * 463\tFalse\n",
      "2318\t7 * 331\tFalse\n",
      "2320\t3 * 773\tFalse\n",
      "2322\t11 * 211\tTrue\n",
      "2324\t23 * 101\tFalse\n",
      "2326\t3 * 5^2 * 31\tFalse\n",
      "2328\t13 * 179\tFalse\n",
      "2330\t17 * 137\tTrue\n",
      "2332\t3^2 * 7 * 37\tFalse\n",
      "2334\t2333\tFalse\n",
      "2336\t5 * 467\tFalse\n",
      "2338\t3 * 19 * 41\tFalse\n",
      "2340\t2339\tTrue\n",
      "2342\t2341\tTrue\n",
      "2344\t3 * 11 * 71\tFalse\n",
      "2346\t5 * 7 * 67\tFalse\n",
      "2348\t2347\tFalse\n",
      "2350\t3^4 * 29\tFalse\n",
      "2352\t2351\tTrue\n",
      "2354\t13 * 181\tFalse\n",
      "2356\t3 * 5 * 157\tFalse\n",
      "2358\t2357\tFalse\n",
      "2360\t7 * 337\tTrue\n",
      "2362\t3 * 787\tFalse\n",
      "2364\t17 * 139\tFalse\n",
      "2366\t5 * 11 * 43\tFalse\n",
      "2368\t3^2 * 263\tFalse\n",
      "2370\t23 * 103\tTrue\n",
      "2372\t2371\tTrue\n",
      "2374\t3 * 7 * 113\tFalse\n",
      "2376\t5^3 * 19\tFalse\n",
      "2378\t2377\tFalse\n",
      "2380\t3 * 13 * 61\tFalse\n",
      "2382\t2381\tTrue\n",
      "2384\t2383\tFalse\n",
      "2386\t3^2 * 5 * 53\tFalse\n",
      "2388\t7 * 11 * 31\tFalse\n",
      "2390\t2389\tTrue\n",
      "2392\t3 * 797\tFalse\n",
      "2394\t2393\tFalse\n",
      "2396\t5 * 479\tFalse\n",
      "2398\t3 * 17 * 47\tFalse\n",
      "2400\t2399\tTrue\n",
      "2402\t7^4\tTrue\n",
      "2404\t3^3 * 89\tFalse\n",
      "2406\t5 * 13 * 37\tFalse\n",
      "2408\t29 * 83\tFalse\n",
      "2410\t3 * 11 * 73\tTrue\n",
      "2412\t2411\tTrue\n",
      "2414\t19 * 127\tFalse\n",
      "2416\t3 * 5 * 7 * 23\tFalse\n",
      "2418\t2417\tFalse\n",
      "2420\t41 * 59\tTrue\n",
      "2422\t3^2 * 269\tFalse\n",
      "2424\t2423\tFalse\n",
      "2426\t5^2 * 97\tFalse\n",
      "2428\t3 * 809\tFalse\n",
      "2430\t7 * 347\tTrue\n",
      "2432\t11 * 13 * 17\tTrue\n",
      "2434\t3 * 811\tFalse\n",
      "2436\t5 * 487\tFalse\n",
      "2438\t2437\tFalse\n",
      "2440\t3^2 * 271\tFalse\n",
      "2442\t2441\tTrue\n",
      "2444\t7 * 349\tFalse\n",
      "2446\t3 * 5 * 163\tFalse\n",
      "2448\t2447\tFalse\n",
      "2450\t31 * 79\tTrue\n",
      "2452\t3 * 19 * 43\tFalse\n",
      "2454\t11 * 223\tFalse\n",
      "2456\t5 * 491\tFalse\n",
      "2458\t3^3 * 7 * 13\tFalse\n",
      "2460\t2459\tTrue\n",
      "2462\t23 * 107\tTrue\n",
      "2464\t3 * 821\tFalse\n",
      "2466\t5 * 17 * 29\tFalse\n",
      "2468\t2467\tFalse\n",
      "2470\t3 * 823\tFalse\n",
      "2472\t7 * 353\tTrue\n",
      "2474\t2473\tFalse\n",
      "2476\t3^2 * 5^2 * 11\tFalse\n",
      "2478\t2477\tFalse\n",
      "2480\t37 * 67\tTrue\n",
      "2482\t3 * 827\tFalse\n",
      "2484\t13 * 191\tFalse\n",
      "2486\t5 * 7 * 71\tFalse\n",
      "2488\t3 * 829\tFalse\n",
      "2490\t19 * 131\tTrue\n",
      "2492\t47 * 53\tTrue\n",
      "2494\t3^2 * 277\tFalse\n",
      "2496\t5 * 499\tFalse\n",
      "2498\t11 * 227\tFalse\n",
      "2500\t3 * 7^2 * 17\tFalse\n",
      "2502\t41 * 61\tTrue\n",
      "2504\t2503\tFalse\n",
      "2506\t3 * 5 * 167\tFalse\n",
      "2508\t23 * 109\tFalse\n",
      "2510\t13 * 193\tTrue\n",
      "2512\t3^4 * 31\tFalse\n",
      "2514\t7 * 359\tFalse\n",
      "2516\t5 * 503\tFalse\n",
      "2518\t3 * 839\tFalse\n",
      "2520\t11 * 229\tTrue\n",
      "2522\t2521\tTrue\n",
      "2524\t3 * 29^2\tFalse\n",
      "2526\t5^2 * 101\tFalse\n",
      "2528\t7 * 19^2\tFalse\n",
      "2530\t3^2 * 281\tFalse\n",
      "2532\t2531\tTrue\n",
      "2534\t17 * 149\tFalse\n",
      "2536\t3 * 5 * 13^2\tFalse\n",
      "2538\t43 * 59\tFalse\n",
      "2540\t2539\tTrue\n",
      "2542\t3 * 7 * 11^2\tFalse\n",
      "2544\t2543\tFalse\n",
      "2546\t5 * 509\tFalse\n",
      "2548\t3^2 * 283\tFalse\n",
      "2550\t2549\tTrue\n",
      "2552\t2551\tTrue\n",
      "2554\t3 * 23 * 37\tFalse\n",
      "2556\t5 * 7 * 73\tFalse\n",
      "2558\t2557\tFalse\n",
      "2560\t3 * 853\tFalse\n",
      "2562\t13 * 197\tTrue\n",
      "2564\t11 * 233\tFalse\n",
      "2566\t3^3 * 5 * 19\tFalse\n",
      "2568\t17 * 151\tFalse\n",
      "2570\t7 * 367\tTrue\n",
      "2572\t3 * 857\tFalse\n",
      "2574\t31 * 83\tFalse\n",
      "2576\t5^2 * 103\tFalse\n",
      "2578\t3 * 859\tFalse\n",
      "2580\t2579\tTrue\n",
      "2582\t29 * 89\tTrue\n",
      "2584\t3^2 * 7 * 41\tFalse\n",
      "2586\t5 * 11 * 47\tFalse\n",
      "2588\t13 * 199\tFalse\n",
      "2590\t3 * 863\tFalse\n",
      "2592\t2591\tTrue\n",
      "2594\t2593\tFalse\n",
      "2596\t3 * 5 * 173\tFalse\n",
      "2598\t7^2 * 53\tFalse\n",
      "2600\t23 * 113\tTrue\n",
      "2602\t3^2 * 17^2\tFalse\n",
      "2604\t19 * 137\tFalse\n",
      "2606\t5 * 521\tFalse\n",
      "2608\t3 * 11 * 79\tFalse\n",
      "2610\t2609\tTrue\n",
      "2612\t7 * 373\tTrue\n",
      "2614\t3 * 13 * 67\tFalse\n",
      "2616\t5 * 523\tFalse\n",
      "2618\t2617\tFalse\n",
      "2620\t3^3 * 97\tFalse\n",
      "2622\t2621\tTrue\n",
      "2624\t43 * 61\tFalse\n",
      "2626\t3 * 5^3 * 7\tFalse\n",
      "2628\t37 * 71\tFalse\n",
      "2630\t11 * 239\tTrue\n",
      "2632\t3 * 877\tFalse\n",
      "2634\t2633\tFalse\n",
      "2636\t5 * 17 * 31\tFalse\n",
      "2638\t3^2 * 293\tFalse\n",
      "2640\t7 * 13 * 29\tTrue\n",
      "2642\t19 * 139\tTrue\n",
      "2644\t3 * 881\tFalse\n",
      "2646\t5 * 23^2\tFalse\n",
      "2648\t2647\tFalse\n",
      "2650\t3 * 883\tFalse\n",
      "2652\t11 * 241\tTrue\n",
      "2654\t7 * 379\tFalse\n",
      "2656\t3^2 * 5 * 59\tFalse\n",
      "2658\t2657\tFalse\n",
      "2660\t2659\tTrue\n",
      "2662\t3 * 887\tFalse\n",
      "2664\t2663\tFalse\n",
      "2666\t5 * 13 * 41\tFalse\n",
      "2668\t3 * 7 * 127\tFalse\n",
      "2670\t17 * 157\tTrue\n",
      "2672\t2671\tTrue\n",
      "2674\t3^5 * 11\tFalse\n",
      "2676\t5^2 * 107\tFalse\n",
      "2678\t2677\tFalse\n",
      "2680\t3 * 19 * 47\tFalse\n",
      "2682\t7 * 383\tTrue\n",
      "2684\t2683\tFalse\n",
      "2686\t3 * 5 * 179\tFalse\n",
      "2688\t2687\tFalse\n",
      "2690\t2689\tTrue\n",
      "2692\t3^2 * 13 * 23\tFalse\n",
      "2694\t2693\tFalse\n",
      "2696\t5 * 7^2 * 11\tFalse\n",
      "2698\t3 * 29 * 31\tFalse\n",
      "2700\t2699\tTrue\n",
      "2702\t37 * 73\tTrue\n",
      "2704\t3 * 17 * 53\tFalse\n",
      "2706\t5 * 541\tFalse\n",
      "2708\t2707\tFalse\n",
      "2710\t3^2 * 7 * 43\tFalse\n",
      "2712\t2711\tTrue\n",
      "2714\t2713\tFalse\n",
      "2716\t3 * 5 * 181\tFalse\n",
      "2718\t11 * 13 * 19\tFalse\n",
      "2720\t2719\tTrue\n",
      "2722\t3 * 907\tFalse\n",
      "2724\t7 * 389\tFalse\n",
      "2726\t5^2 * 109\tFalse\n",
      "2728\t3^3 * 101\tFalse\n",
      "2730\t2729\tTrue\n",
      "2732\t2731\tTrue\n",
      "2734\t3 * 911\tFalse\n",
      "2736\t5 * 547\tFalse\n",
      "2738\t7 * 17 * 23\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2740\t3 * 11 * 83\tTrue\n",
      "2742\t2741\tTrue\n",
      "2744\t13 * 211\tFalse\n",
      "2746\t3^2 * 5 * 61\tFalse\n",
      "2748\t41 * 67\tFalse\n",
      "2750\t2749\tTrue\n",
      "2752\t3 * 7 * 131\tFalse\n",
      "2754\t2753\tFalse\n",
      "2756\t5 * 19 * 29\tFalse\n",
      "2758\t3 * 919\tFalse\n",
      "2760\t31 * 89\tTrue\n",
      "2762\t11 * 251\tTrue\n",
      "2764\t3^2 * 307\tFalse\n",
      "2766\t5 * 7 * 79\tFalse\n",
      "2768\t2767\tFalse\n",
      "2770\t3 * 13 * 71\tTrue\n",
      "2772\t17 * 163\tTrue\n",
      "2774\t47 * 59\tFalse\n",
      "2776\t3 * 5^2 * 37\tFalse\n",
      "2778\t2777\tFalse\n",
      "2780\t7 * 397\tTrue\n",
      "2782\t3^3 * 103\tFalse\n",
      "2784\t11^2 * 23\tFalse\n",
      "2786\t5 * 557\tFalse\n",
      "2788\t3 * 929\tFalse\n",
      "2790\t2789\tTrue\n",
      "2792\t2791\tTrue\n",
      "2794\t3 * 7^2 * 19\tFalse\n",
      "2796\t5 * 13 * 43\tFalse\n",
      "2798\t2797\tFalse\n",
      "2800\t3^2 * 311\tFalse\n",
      "2802\t2801\tTrue\n",
      "2804\t2803\tFalse\n",
      "2806\t3 * 5 * 11 * 17\tFalse\n",
      "2808\t7 * 401\tFalse\n",
      "2810\t53^2\tTrue\n",
      "2812\t3 * 937\tFalse\n",
      "2814\t29 * 97\tFalse\n",
      "2816\t5 * 563\tFalse\n",
      "2818\t3^2 * 313\tFalse\n",
      "2820\t2819\tTrue\n",
      "2822\t7 * 13 * 31\tTrue\n",
      "2824\t3 * 941\tFalse\n",
      "2826\t5^2 * 113\tFalse\n",
      "2828\t11 * 257\tFalse\n",
      "2830\t3 * 23 * 41\tFalse\n",
      "2832\t19 * 149\tTrue\n",
      "2834\t2833\tFalse\n",
      "2836\t3^4 * 5 * 7\tFalse\n",
      "2838\t2837\tFalse\n",
      "2840\t17 * 167\tTrue\n",
      "2842\t3 * 947\tFalse\n",
      "2844\t2843\tFalse\n",
      "2846\t5 * 569\tFalse\n",
      "2848\t3 * 13 * 73\tFalse\n",
      "2850\t7 * 11 * 37\tTrue\n",
      "2852\t2851\tTrue\n",
      "2854\t3^2 * 317\tFalse\n",
      "2856\t5 * 571\tFalse\n",
      "2858\t2857\tFalse\n",
      "2860\t3 * 953\tFalse\n",
      "2862\t2861\tTrue\n",
      "2864\t7 * 409\tFalse\n",
      "2866\t3 * 5 * 191\tFalse\n",
      "2868\t47 * 61\tFalse\n",
      "2870\t19 * 151\tTrue\n",
      "2872\t3^2 * 11 * 29\tTrue\n",
      "2874\t13^2 * 17\tFalse\n",
      "2876\t5^3 * 23\tFalse\n",
      "2878\t3 * 7 * 137\tFalse\n",
      "2880\t2879\tTrue\n",
      "2882\t43 * 67\tTrue\n",
      "2884\t3 * 31^2\tFalse\n",
      "2886\t5 * 577\tFalse\n",
      "2888\t2887\tFalse\n",
      "2890\t3^3 * 107\tFalse\n",
      "2892\t7^2 * 59\tTrue\n",
      "2894\t11 * 263\tFalse\n",
      "2896\t3 * 5 * 193\tFalse\n",
      "2898\t2897\tFalse\n",
      "2900\t13 * 223\tTrue\n",
      "2902\t3 * 967\tFalse\n",
      "2904\t2903\tFalse\n",
      "2906\t5 * 7 * 83\tFalse\n",
      "2908\t3^2 * 17 * 19\tFalse\n",
      "2910\t2909\tTrue\n",
      "2912\t41 * 71\tTrue\n",
      "2914\t3 * 971\tFalse\n",
      "2916\t5 * 11 * 53\tFalse\n",
      "2918\t2917\tFalse\n",
      "2920\t3 * 7 * 139\tFalse\n",
      "2922\t23 * 127\tTrue\n",
      "2924\t37 * 79\tFalse\n",
      "2926\t3^2 * 5^2 * 13\tFalse\n",
      "2928\t2927\tFalse\n",
      "2930\t29 * 101\tTrue\n",
      "2932\t3 * 977\tFalse\n",
      "2934\t7 * 419\tFalse\n",
      "2936\t5 * 587\tFalse\n",
      "2938\t3 * 11 * 89\tFalse\n",
      "2940\t2939\tTrue\n",
      "2942\t17 * 173\tTrue\n",
      "2944\t3^3 * 109\tFalse\n",
      "2946\t5 * 19 * 31\tFalse\n",
      "2948\t7 * 421\tFalse\n",
      "2950\t3 * 983\tFalse\n",
      "2952\t13 * 227\tTrue\n",
      "2954\t2953\tFalse\n",
      "2956\t3 * 5 * 197\tFalse\n",
      "2958\t2957\tFalse\n",
      "2960\t11 * 269\tTrue\n",
      "2962\t3^2 * 7 * 47\tFalse\n",
      "2964\t2963\tFalse\n",
      "2966\t5 * 593\tFalse\n",
      "2968\t3 * 23 * 43\tFalse\n",
      "2970\t2969\tTrue\n",
      "2972\t2971\tTrue\n",
      "2974\t3 * 991\tFalse\n",
      "2976\t5^2 * 7 * 17\tFalse\n",
      "2978\t13 * 229\tFalse\n",
      "2980\t3^2 * 331\tFalse\n",
      "2982\t11 * 271\tTrue\n",
      "2984\t19 * 157\tFalse\n",
      "2986\t3 * 5 * 199\tFalse\n",
      "2988\t29 * 103\tFalse\n",
      "2990\t7^2 * 61\tTrue\n",
      "2992\t3 * 997\tFalse\n",
      "2994\t41 * 73\tFalse\n",
      "2996\t5 * 599\tFalse\n",
      "2998\t3^4 * 37\tFalse\n",
      "3000\t2999\tTrue\n",
      "3002\t3001\tTrue\n",
      "3004\t3 * 7 * 11 * 13\tFalse\n",
      "3006\t5 * 601\tFalse\n",
      "3008\t31 * 97\tFalse\n",
      "3010\t3 * 17 * 59\tTrue\n",
      "3012\t3011\tTrue\n",
      "3014\t23 * 131\tFalse\n",
      "3016\t3^2 * 5 * 67\tFalse\n",
      "3018\t7 * 431\tFalse\n",
      "3020\t3019\tTrue\n",
      "3022\t3 * 19 * 53\tTrue\n",
      "3024\t3023\tFalse\n",
      "3026\t5^2 * 11^2\tFalse\n",
      "3028\t3 * 1009\tFalse\n",
      "3030\t13 * 233\tTrue\n",
      "3032\t7 * 433\tTrue\n",
      "3034\t3^2 * 337\tFalse\n",
      "3036\t5 * 607\tFalse\n",
      "3038\t3037\tFalse\n",
      "3040\t3 * 1013\tFalse\n",
      "3042\t3041\tTrue\n",
      "3044\t17 * 179\tFalse\n",
      "3046\t3 * 5 * 7 * 29\tFalse\n",
      "3048\t11 * 277\tFalse\n",
      "3050\t3049\tTrue\n",
      "3052\t3^3 * 113\tFalse\n",
      "3054\t43 * 71\tFalse\n",
      "3056\t5 * 13 * 47\tFalse\n",
      "3058\t3 * 1019\tFalse\n",
      "3060\t7 * 19 * 23\tTrue\n",
      "3062\t3061\tTrue\n",
      "3064\t3 * 1021\tFalse\n",
      "3066\t5 * 613\tFalse\n",
      "3068\t3067\tFalse\n",
      "3070\t3^2 * 11 * 31\tTrue\n",
      "3072\t37 * 83\tTrue\n",
      "3074\t7 * 439\tFalse\n",
      "3076\t3 * 5^2 * 41\tFalse\n",
      "3078\t17 * 181\tFalse\n",
      "3080\t3079\tTrue\n",
      "3082\t3 * 13 * 79\tTrue\n",
      "3084\t3083\tFalse\n",
      "3086\t5 * 617\tFalse\n",
      "3088\t3^2 * 7^3\tFalse\n",
      "3090\t3089\tTrue\n",
      "3092\t11 * 281\tTrue\n",
      "3094\t3 * 1031\tFalse\n",
      "3096\t5 * 619\tFalse\n",
      "3098\t19 * 163\tFalse\n",
      "3100\t3 * 1033\tFalse\n",
      "3102\t7 * 443\tTrue\n",
      "3104\t29 * 107\tFalse\n",
      "3106\t3^3 * 5 * 23\tFalse\n",
      "3108\t13 * 239\tFalse\n",
      "3110\t3109\tTrue\n",
      "3112\t3 * 17 * 61\tFalse\n",
      "3114\t11 * 283\tFalse\n",
      "3116\t5 * 7 * 89\tFalse\n",
      "3118\t3 * 1039\tFalse\n",
      "3120\t3119\tTrue\n",
      "3122\t3121\tTrue\n",
      "3124\t3^2 * 347\tFalse\n",
      "3126\t5^5\tFalse\n",
      "3128\t53 * 59\tFalse\n",
      "3130\t3 * 7 * 149\tFalse\n",
      "3132\t31 * 101\tTrue\n",
      "3134\t13 * 241\tFalse\n",
      "3136\t3 * 5 * 11 * 19\tFalse\n",
      "3138\t3137\tFalse\n",
      "3140\t43 * 73\tTrue\n",
      "3142\t3^2 * 349\tFalse\n",
      "3144\t7 * 449\tFalse\n",
      "3146\t5 * 17 * 37\tFalse\n",
      "3148\t3 * 1049\tFalse\n",
      "3150\t47 * 67\tTrue\n",
      "3152\t23 * 137\tTrue\n",
      "3154\t3 * 1051\tFalse\n",
      "3156\t5 * 631\tFalse\n",
      "3158\t7 * 11 * 41\tFalse\n",
      "3160\t3^5 * 13\tFalse\n",
      "3162\t29 * 109\tTrue\n",
      "3164\t3163\tFalse\n",
      "3166\t3 * 5 * 211\tFalse\n",
      "3168\t3167\tFalse\n",
      "3170\t3169\tTrue\n",
      "3172\t3 * 7 * 151\tFalse\n",
      "3174\t19 * 167\tFalse\n",
      "3176\t5^2 * 127\tFalse\n",
      "3178\t3^2 * 353\tFalse\n",
      "3180\t11 * 17^2\tTrue\n",
      "3182\t3181\tTrue\n",
      "3184\t3 * 1061\tFalse\n",
      "3186\t5 * 7^2 * 13\tFalse\n",
      "3188\t3187\tFalse\n",
      "3190\t3 * 1063\tFalse\n",
      "3192\t3191\tTrue\n",
      "3194\t31 * 103\tFalse\n",
      "3196\t3^2 * 5 * 71\tFalse\n",
      "3198\t23 * 139\tFalse\n",
      "3200\t7 * 457\tTrue\n",
      "3202\t3 * 11 * 97\tTrue\n",
      "3204\t3203\tFalse\n",
      "3206\t5 * 641\tFalse\n",
      "3208\t3 * 1069\tFalse\n",
      "3210\t3209\tTrue\n",
      "3212\t13^2 * 19\tTrue\n",
      "3214\t3^3 * 7 * 17\tFalse\n",
      "3216\t5 * 643\tFalse\n",
      "3218\t3217\tFalse\n",
      "3220\t3 * 29 * 37\tFalse\n",
      "3222\t3221\tTrue\n",
      "3224\t11 * 293\tFalse\n",
      "3226\t3 * 5^2 * 43\tFalse\n",
      "3228\t7 * 461\tFalse\n",
      "3230\t3229\tTrue\n",
      "3232\t3^2 * 359\tFalse\n",
      "3234\t53 * 61\tFalse\n",
      "3236\t5 * 647\tFalse\n",
      "3238\t3 * 13 * 83\tFalse\n",
      "3240\t41 * 79\tTrue\n",
      "3242\t7 * 463\tTrue\n",
      "3244\t3 * 23 * 47\tFalse\n",
      "3246\t5 * 11 * 59\tFalse\n",
      "3248\t17 * 191\tFalse\n",
      "3250\t3^2 * 19^2\tFalse\n",
      "3252\t3251\tTrue\n",
      "3254\t3253\tFalse\n",
      "3256\t3 * 5 * 7 * 31\tFalse\n",
      "3258\t3257\tFalse\n",
      "3260\t3259\tTrue\n",
      "3262\t3 * 1087\tFalse\n",
      "3264\t13 * 251\tFalse\n",
      "3266\t5 * 653\tFalse\n",
      "3268\t3^3 * 11^2\tFalse\n",
      "3270\t7 * 467\tTrue\n",
      "3272\t3271\tTrue\n",
      "3274\t3 * 1091\tFalse\n",
      "3276\t5^2 * 131\tFalse\n",
      "3278\t29 * 113\tFalse\n",
      "3280\t3 * 1093\tFalse\n",
      "3282\t17 * 193\tTrue\n",
      "3284\t7^2 * 67\tFalse\n",
      "3286\t3^2 * 5 * 73\tFalse\n",
      "3288\t19 * 173\tFalse\n",
      "3290\t11 * 13 * 23\tTrue\n",
      "3292\t3 * 1097\tFalse\n",
      "3294\t37 * 89\tFalse\n",
      "3296\t5 * 659\tFalse\n",
      "3298\t3 * 7 * 157\tFalse\n",
      "3300\t3299\tTrue\n",
      "3302\t3301\tTrue\n",
      "3304\t3^2 * 367\tFalse\n",
      "3306\t5 * 661\tFalse\n",
      "3308\t3307\tFalse\n",
      "3310\t3 * 1103\tFalse\n",
      "3312\t7 * 11 * 43\tTrue\n",
      "3314\t3313\tFalse\n",
      "3316\t3 * 5 * 13 * 17\tFalse\n",
      "3318\t31 * 107\tFalse\n",
      "3320\t3319\tTrue\n",
      "3322\t3^4 * 41\tFalse\n",
      "3324\t3323\tFalse\n",
      "3326\t5^2 * 7 * 19\tFalse\n",
      "3328\t3 * 1109\tFalse\n",
      "3330\t3329\tTrue\n",
      "3332\t3331\tTrue\n",
      "3334\t3 * 11 * 101\tFalse\n",
      "3336\t5 * 23 * 29\tFalse\n",
      "3338\t47 * 71\tFalse\n",
      "3340\t3^2 * 7 * 53\tFalse\n",
      "3342\t13 * 257\tTrue\n",
      "3344\t3343\tFalse\n",
      "3346\t3 * 5 * 223\tFalse\n",
      "3348\t3347\tFalse\n",
      "3350\t17 * 197\tTrue\n",
      "3352\t3 * 1117\tFalse\n",
      "3354\t7 * 479\tFalse\n",
      "3356\t5 * 11 * 61\tFalse\n",
      "3358\t3^2 * 373\tFalse\n",
      "3360\t3359\tTrue\n",
      "3362\t3361\tTrue\n",
      "3364\t3 * 19 * 59\tFalse\n",
      "3366\t5 * 673\tFalse\n",
      "3368\t7 * 13 * 37\tFalse\n",
      "3370\t3 * 1123\tFalse\n",
      "3372\t3371\tTrue\n",
      "3374\t3373\tFalse\n",
      "3376\t3^3 * 5^3\tFalse\n",
      "3378\t11 * 307\tFalse\n",
      "3380\t31 * 109\tTrue\n",
      "3382\t3 * 7^2 * 23\tFalse\n",
      "3384\t17 * 199\tFalse\n",
      "3386\t5 * 677\tFalse\n",
      "3388\t3 * 1129\tFalse\n",
      "3390\t3389\tTrue\n",
      "3392\t3391\tTrue\n",
      "3394\t3^2 * 13 * 29\tFalse\n",
      "3396\t5 * 7 * 97\tFalse\n",
      "3398\t43 * 79\tFalse\n",
      "3400\t3 * 11 * 103\tTrue\n",
      "3402\t19 * 179\tTrue\n",
      "3404\t41 * 83\tFalse\n",
      "3406\t3 * 5 * 227\tFalse\n",
      "3408\t3407\tFalse\n",
      "3410\t7 * 487\tTrue\n",
      "3412\t3^2 * 379\tFalse\n",
      "3414\t3413\tFalse\n",
      "3416\t5 * 683\tFalse\n",
      "3418\t3 * 17 * 67\tFalse\n",
      "3420\t13 * 263\tTrue\n",
      "3422\t11 * 311\tTrue\n",
      "3424\t3 * 7 * 163\tFalse\n",
      "3426\t5^2 * 137\tFalse\n",
      "3428\t23 * 149\tFalse\n",
      "3430\t3^3 * 127\tFalse\n",
      "3432\t47 * 73\tTrue\n",
      "3434\t3433\tFalse\n",
      "3436\t3 * 5 * 229\tFalse\n",
      "3438\t7 * 491\tFalse\n",
      "3440\t19 * 181\tTrue\n",
      "3442\t3 * 31 * 37\tTrue\n",
      "3444\t11 * 313\tFalse\n",
      "3446\t5 * 13 * 53\tFalse\n",
      "3448\t3^2 * 383\tFalse\n",
      "3450\t3449\tTrue\n",
      "3452\t7 * 17 * 29\tTrue\n",
      "3454\t3 * 1151\tFalse\n",
      "3456\t5 * 691\tFalse\n",
      "3458\t3457\tFalse\n",
      "3460\t3 * 1153\tFalse\n",
      "3462\t3461\tTrue\n",
      "3464\t3463\tFalse\n",
      "3466\t3^2 * 5 * 7 * 11\tFalse\n",
      "3468\t3467\tFalse\n",
      "3470\t3469\tTrue\n",
      "3472\t3 * 13 * 89\tFalse\n",
      "3474\t23 * 151\tFalse\n",
      "3476\t5^2 * 139\tFalse\n",
      "3478\t3 * 19 * 61\tFalse\n",
      "3480\t7^2 * 71\tTrue\n",
      "3482\t59^2\tTrue\n",
      "3484\t3^4 * 43\tFalse\n",
      "3486\t5 * 17 * 41\tFalse\n",
      "3488\t11 * 317\tFalse\n",
      "3490\t3 * 1163\tFalse\n",
      "3492\t3491\tTrue\n",
      "3494\t7 * 499\tFalse\n",
      "3496\t3 * 5 * 233\tFalse\n",
      "3498\t13 * 269\tFalse\n",
      "3500\t3499\tTrue\n",
      "3502\t3^2 * 389\tFalse\n",
      "3504\t31 * 113\tFalse\n",
      "3506\t5 * 701\tFalse\n",
      "3508\t3 * 7 * 167\tFalse\n",
      "3510\t11^2 * 29\tTrue\n",
      "3512\t3511\tTrue\n",
      "3514\t3 * 1171\tFalse\n",
      "3516\t5 * 19 * 37\tFalse\n",
      "3518\t3517\tFalse\n",
      "3520\t3^2 * 17 * 23\tFalse\n",
      "3522\t7 * 503\tTrue\n",
      "3524\t13 * 271\tFalse\n",
      "3526\t3 * 5^2 * 47\tFalse\n",
      "3528\t3527\tFalse\n",
      "3530\t3529\tTrue\n",
      "3532\t3 * 11 * 107\tTrue\n",
      "3534\t3533\tFalse\n",
      "3536\t5 * 7 * 101\tFalse\n",
      "3538\t3^3 * 131\tFalse\n",
      "3540\t3539\tTrue\n",
      "3542\t3541\tTrue\n",
      "3544\t3 * 1181\tFalse\n",
      "3546\t5 * 709\tFalse\n",
      "3548\t3547\tFalse\n",
      "3550\t3 * 7 * 13^2\tFalse\n",
      "3552\t53 * 67\tTrue\n",
      "3554\t11 * 17 * 19\tFalse\n",
      "3556\t3^2 * 5 * 79\tFalse\n",
      "3558\t3557\tFalse\n",
      "3560\t3559\tTrue\n",
      "3562\t3 * 1187\tFalse\n",
      "3564\t7 * 509\tFalse\n",
      "3566\t5 * 23 * 31\tFalse\n",
      "3568\t3 * 29 * 41\tFalse\n",
      "3570\t43 * 83\tTrue\n",
      "3572\t3571\tTrue\n",
      "3574\t3^2 * 397\tFalse\n",
      "3576\t5^2 * 11 * 13\tFalse\n",
      "3578\t7^2 * 73\tFalse\n",
      "3580\t3 * 1193\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3582\t3581\tTrue\n",
      "3584\t3583\tFalse\n",
      "3586\t3 * 5 * 239\tFalse\n",
      "3588\t17 * 211\tFalse\n",
      "3590\t37 * 97\tTrue\n",
      "3592\t3^3 * 7 * 19\tFalse\n",
      "3594\t3593\tFalse\n",
      "3596\t5 * 719\tFalse\n",
      "3598\t3 * 11 * 109\tFalse\n",
      "3600\t59 * 61\tTrue\n",
      "3602\t13 * 277\tTrue\n",
      "3604\t3 * 1201\tFalse\n",
      "3606\t5 * 7 * 103\tFalse\n",
      "3608\t3607\tFalse\n",
      "3610\t3^2 * 401\tFalse\n",
      "3612\t23 * 157\tTrue\n",
      "3614\t3613\tFalse\n",
      "3616\t3 * 5 * 241\tFalse\n",
      "3618\t3617\tFalse\n",
      "3620\t7 * 11 * 47\tTrue\n",
      "3622\t3 * 17 * 71\tTrue\n",
      "3624\t3623\tFalse\n",
      "3626\t5^3 * 29\tFalse\n",
      "3628\t3^2 * 13 * 31\tFalse\n",
      "3630\t19 * 191\tTrue\n",
      "3632\t3631\tTrue\n",
      "3634\t3 * 7 * 173\tFalse\n",
      "3636\t5 * 727\tFalse\n",
      "3638\t3637\tFalse\n",
      "3640\t3 * 1213\tFalse\n",
      "3642\t11 * 331\tTrue\n",
      "3644\t3643\tFalse\n",
      "3646\t3^6 * 5\tFalse\n",
      "3648\t7 * 521\tFalse\n",
      "3650\t41 * 89\tTrue\n",
      "3652\t3 * 1217\tFalse\n",
      "3654\t13 * 281\tFalse\n",
      "3656\t5 * 17 * 43\tFalse\n",
      "3658\t3 * 23 * 53\tFalse\n",
      "3660\t3659\tTrue\n",
      "3662\t7 * 523\tTrue\n",
      "3664\t3^2 * 11 * 37\tFalse\n",
      "3666\t5 * 733\tFalse\n",
      "3668\t19 * 193\tFalse\n",
      "3670\t3 * 1223\tFalse\n",
      "3672\t3671\tTrue\n",
      "3674\t3673\tFalse\n",
      "3676\t3 * 5^2 * 7^2\tFalse\n",
      "3678\t3677\tFalse\n",
      "3680\t13 * 283\tTrue\n",
      "3682\t3^2 * 409\tFalse\n",
      "3684\t29 * 127\tFalse\n",
      "3686\t5 * 11 * 67\tFalse\n",
      "3688\t3 * 1229\tFalse\n",
      "3690\t7 * 17 * 31\tTrue\n",
      "3692\t3691\tTrue\n",
      "3694\t3 * 1231\tFalse\n",
      "3696\t5 * 739\tFalse\n",
      "3698\t3697\tFalse\n",
      "3700\t3^3 * 137\tFalse\n",
      "3702\t3701\tTrue\n",
      "3704\t7 * 23^2\tFalse\n",
      "3706\t3 * 5 * 13 * 19\tFalse\n",
      "3708\t11 * 337\tFalse\n",
      "3710\t3709\tTrue\n",
      "3712\t3 * 1237\tFalse\n",
      "3714\t47 * 79\tFalse\n",
      "3716\t5 * 743\tFalse\n",
      "3718\t3^2 * 7 * 59\tFalse\n",
      "3720\t3719\tTrue\n",
      "3722\t61^2\tTrue\n",
      "3724\t3 * 17 * 73\tFalse\n",
      "3726\t5^2 * 149\tFalse\n",
      "3728\t3727\tFalse\n",
      "3730\t3 * 11 * 113\tTrue\n",
      "3732\t7 * 13 * 41\tTrue\n",
      "3734\t3733\tFalse\n",
      "3736\t3^2 * 5 * 83\tFalse\n",
      "3738\t37 * 101\tFalse\n",
      "3740\t3739\tTrue\n",
      "3742\t3 * 29 * 43\tFalse\n",
      "3744\t19 * 197\tFalse\n",
      "3746\t5 * 7 * 107\tFalse\n",
      "3748\t3 * 1249\tFalse\n",
      "3750\t23 * 163\tTrue\n",
      "3752\t11^2 * 31\tTrue\n",
      "3754\t3^3 * 139\tFalse\n",
      "3756\t5 * 751\tFalse\n",
      "3758\t13 * 17^2\tFalse\n",
      "3760\t3 * 7 * 179\tFalse\n",
      "3762\t3761\tTrue\n",
      "3764\t53 * 71\tFalse\n",
      "3766\t3 * 5 * 251\tFalse\n",
      "3768\t3767\tFalse\n",
      "3770\t3769\tTrue\n",
      "3772\t3^2 * 419\tFalse\n",
      "3774\t7^3 * 11\tFalse\n",
      "3776\t5^2 * 151\tFalse\n",
      "3778\t3 * 1259\tFalse\n",
      "3780\t3779\tTrue\n",
      "3782\t19 * 199\tTrue\n",
      "3784\t3 * 13 * 97\tFalse\n",
      "3786\t5 * 757\tFalse\n",
      "3788\t7 * 541\tFalse\n",
      "3790\t3^2 * 421\tFalse\n",
      "3792\t17 * 223\tTrue\n",
      "3794\t3793\tFalse\n",
      "3796\t3 * 5 * 11 * 23\tFalse\n",
      "3798\t3797\tFalse\n",
      "3800\t29 * 131\tTrue\n",
      "3802\t3 * 7 * 181\tFalse\n",
      "3804\t3803\tFalse\n",
      "3806\t5 * 761\tFalse\n",
      "3808\t3^4 * 47\tFalse\n",
      "3810\t13 * 293\tTrue\n",
      "3812\t37 * 103\tTrue\n",
      "3814\t3 * 31 * 41\tFalse\n",
      "3816\t5 * 7 * 109\tFalse\n",
      "3818\t11 * 347\tFalse\n",
      "3820\t3 * 19 * 67\tFalse\n",
      "3822\t3821\tTrue\n",
      "3824\t3823\tFalse\n",
      "3826\t3^2 * 5^2 * 17\tFalse\n",
      "3828\t43 * 89\tFalse\n",
      "3830\t7 * 547\tTrue\n",
      "3832\t3 * 1277\tFalse\n",
      "3834\t3833\tFalse\n",
      "3836\t5 * 13 * 59\tFalse\n",
      "3838\t3 * 1279\tFalse\n",
      "3840\t11 * 349\tTrue\n",
      "3842\t23 * 167\tTrue\n",
      "3844\t3^2 * 7 * 61\tFalse\n",
      "3846\t5 * 769\tFalse\n",
      "3848\t3847\tFalse\n",
      "3850\t3 * 1283\tFalse\n",
      "3852\t3851\tTrue\n",
      "3854\t3853\tFalse\n",
      "3856\t3 * 5 * 257\tFalse\n",
      "3858\t7 * 19 * 29\tFalse\n",
      "3860\t17 * 227\tTrue\n",
      "3862\t3^3 * 11 * 13\tTrue\n",
      "3864\t3863\tFalse\n",
      "3866\t5 * 773\tFalse\n",
      "3868\t3 * 1289\tFalse\n",
      "3870\t53 * 73\tTrue\n",
      "3872\t7^2 * 79\tTrue\n",
      "3874\t3 * 1291\tFalse\n",
      "3876\t5^3 * 31\tFalse\n",
      "3878\t3877\tFalse\n",
      "3880\t3^2 * 431\tFalse\n",
      "3882\t3881\tTrue\n",
      "3884\t11 * 353\tFalse\n",
      "3886\t3 * 5 * 7 * 37\tFalse\n",
      "3888\t13^2 * 23\tFalse\n",
      "3890\t3889\tTrue\n",
      "3892\t3 * 1297\tFalse\n",
      "3894\t17 * 229\tFalse\n",
      "3896\t5 * 19 * 41\tFalse\n",
      "3898\t3^2 * 433\tFalse\n",
      "3900\t7 * 557\tTrue\n",
      "3902\t47 * 83\tTrue\n",
      "3904\t3 * 1301\tFalse\n",
      "3906\t5 * 11 * 71\tFalse\n",
      "3908\t3907\tFalse\n",
      "3910\t3 * 1303\tFalse\n",
      "3912\t3911\tTrue\n",
      "3914\t7 * 13 * 43\tFalse\n",
      "3916\t3^3 * 5 * 29\tFalse\n",
      "3918\t3917\tFalse\n",
      "3920\t3919\tTrue\n",
      "3922\t3 * 1307\tFalse\n",
      "3924\t3923\tFalse\n",
      "3926\t5^2 * 157\tFalse\n",
      "3928\t3 * 7 * 11 * 17\tFalse\n",
      "3930\t3929\tTrue\n",
      "3932\t3931\tTrue\n",
      "3934\t3^2 * 19 * 23\tFalse\n",
      "3936\t5 * 787\tFalse\n",
      "3938\t31 * 127\tFalse\n",
      "3940\t3 * 13 * 101\tTrue\n",
      "3942\t7 * 563\tTrue\n",
      "3944\t3943\tFalse\n",
      "3946\t3 * 5 * 263\tFalse\n",
      "3948\t3947\tFalse\n",
      "3950\t11 * 359\tTrue\n",
      "3952\t3^2 * 439\tFalse\n",
      "3954\t59 * 67\tFalse\n",
      "3956\t5 * 7 * 113\tFalse\n",
      "3958\t3 * 1319\tFalse\n",
      "3960\t37 * 107\tTrue\n",
      "3962\t17 * 233\tTrue\n",
      "3964\t3 * 1321\tFalse\n",
      "3966\t5 * 13 * 61\tFalse\n",
      "3968\t3967\tFalse\n",
      "3970\t3^4 * 7^2\tFalse\n",
      "3972\t11 * 19^2\tTrue\n",
      "3974\t29 * 137\tFalse\n",
      "3976\t3 * 5^2 * 53\tFalse\n",
      "3978\t41 * 97\tFalse\n",
      "3980\t23 * 173\tTrue\n",
      "3982\t3 * 1327\tFalse\n",
      "3984\t7 * 569\tFalse\n",
      "3986\t5 * 797\tFalse\n",
      "3988\t3^2 * 443\tFalse\n",
      "3990\t3989\tTrue\n",
      "3992\t13 * 307\tTrue\n",
      "3994\t3 * 11^3\tFalse\n",
      "3996\t5 * 17 * 47\tFalse\n",
      "3998\t7 * 571\tFalse\n",
      "4000\t3 * 31 * 43\tFalse\n",
      "4002\t4001\tTrue\n",
      "4004\t4003\tFalse\n",
      "4006\t3^2 * 5 * 89\tFalse\n",
      "4008\t4007\tFalse\n",
      "4010\t19 * 211\tTrue\n",
      "4012\t3 * 7 * 191\tFalse\n",
      "4014\t4013\tFalse\n",
      "4016\t5 * 11 * 73\tFalse\n",
      "4018\t3 * 13 * 103\tFalse\n",
      "4020\t4019\tTrue\n",
      "4022\t4021\tTrue\n",
      "4024\t3^3 * 149\tFalse\n",
      "4026\t5^2 * 7 * 23\tFalse\n",
      "4028\t4027\tFalse\n",
      "4030\t3 * 17 * 79\tTrue\n",
      "4032\t29 * 139\tTrue\n",
      "4034\t37 * 109\tFalse\n",
      "4036\t3 * 5 * 269\tFalse\n",
      "4038\t11 * 367\tFalse\n",
      "4040\t7 * 577\tTrue\n",
      "4042\t3^2 * 449\tFalse\n",
      "4044\t13 * 311\tFalse\n",
      "4046\t5 * 809\tFalse\n",
      "4048\t3 * 19 * 71\tFalse\n",
      "4050\t4049\tTrue\n",
      "4052\t4051\tTrue\n",
      "4054\t3 * 7 * 193\tFalse\n",
      "4056\t5 * 811\tFalse\n",
      "4058\t4057\tFalse\n",
      "4060\t3^2 * 11 * 41\tTrue\n",
      "4062\t31 * 131\tTrue\n",
      "4064\t17 * 239\tFalse\n",
      "4066\t3 * 5 * 271\tFalse\n",
      "4068\t7^2 * 83\tFalse\n",
      "4070\t13 * 313\tTrue\n",
      "4072\t3 * 23 * 59\tFalse\n",
      "4074\t4073\tFalse\n",
      "4076\t5^2 * 163\tFalse\n",
      "4078\t3^3 * 151\tFalse\n",
      "4080\t4079\tTrue\n",
      "4082\t7 * 11 * 53\tTrue\n",
      "4084\t3 * 1361\tFalse\n",
      "4086\t5 * 19 * 43\tFalse\n",
      "4088\t61 * 67\tFalse\n",
      "4090\t3 * 29 * 47\tFalse\n",
      "4092\t4091\tTrue\n",
      "4094\t4093\tFalse\n",
      "4096\t3^2 * 5 * 7 * 13\tFalse\n",
      "4098\t17 * 241\tFalse\n",
      "4100\t4099\tTrue\n",
      "4102\t3 * 1367\tFalse\n",
      "4104\t11 * 373\tFalse\n",
      "4106\t5 * 821\tFalse\n",
      "4108\t3 * 37^2\tFalse\n",
      "4110\t7 * 587\tTrue\n",
      "4112\t4111\tTrue\n",
      "4114\t3^2 * 457\tFalse\n",
      "4116\t5 * 823\tFalse\n",
      "4118\t23 * 179\tFalse\n",
      "4120\t3 * 1373\tFalse\n",
      "4122\t13 * 317\tTrue\n",
      "4124\t7 * 19 * 31\tFalse\n",
      "4126\t3 * 5^3 * 11\tFalse\n",
      "4128\t4127\tFalse\n",
      "4130\t4129\tTrue\n",
      "4132\t3^5 * 17\tFalse\n",
      "4134\t4133\tFalse\n",
      "4136\t5 * 827\tFalse\n",
      "4138\t3 * 7 * 197\tFalse\n",
      "4140\t4139\tTrue\n",
      "4142\t41 * 101\tTrue\n",
      "4144\t3 * 1381\tFalse\n",
      "4146\t5 * 829\tFalse\n",
      "4148\t11 * 13 * 29\tFalse\n",
      "4150\t3^2 * 461\tFalse\n",
      "4152\t7 * 593\tTrue\n",
      "4154\t4153\tFalse\n",
      "4156\t3 * 5 * 277\tFalse\n",
      "4158\t4157\tFalse\n",
      "4160\t4159\tTrue\n",
      "4162\t3 * 19 * 73\tTrue\n",
      "4164\t23 * 181\tFalse\n",
      "4166\t5 * 7^2 * 17\tFalse\n",
      "4168\t3^2 * 463\tFalse\n",
      "4170\t11 * 379\tTrue\n",
      "4172\t43 * 97\tTrue\n",
      "4174\t3 * 13 * 107\tFalse\n",
      "4176\t5^2 * 167\tFalse\n",
      "4178\t4177\tFalse\n",
      "4180\t3 * 7 * 199\tFalse\n",
      "4182\t37 * 113\tTrue\n",
      "4184\t47 * 89\tFalse\n",
      "4186\t3^3 * 5 * 31\tFalse\n",
      "4188\t53 * 79\tFalse\n",
      "4190\t59 * 71\tTrue\n",
      "4192\t3 * 11 * 127\tTrue\n",
      "4194\t7 * 599\tFalse\n",
      "4196\t5 * 839\tFalse\n",
      "4198\t3 * 1399\tFalse\n",
      "4200\t13 * 17 * 19\tTrue\n",
      "4202\t4201\tTrue\n",
      "4204\t3^2 * 467\tFalse\n",
      "4206\t5 * 29^2\tFalse\n",
      "4208\t7 * 601\tFalse\n",
      "4210\t3 * 23 * 61\tFalse\n",
      "4212\t4211\tTrue\n",
      "4214\t11 * 383\tFalse\n",
      "4216\t3 * 5 * 281\tFalse\n",
      "4218\t4217\tFalse\n",
      "4220\t4219\tTrue\n",
      "4222\t3^2 * 7 * 67\tFalse\n",
      "4224\t41 * 103\tFalse\n",
      "4226\t5^2 * 13^2\tFalse\n",
      "4228\t3 * 1409\tFalse\n",
      "4230\t4229\tTrue\n",
      "4232\t4231\tTrue\n",
      "4234\t3 * 17 * 83\tFalse\n",
      "4236\t5 * 7 * 11^2\tFalse\n",
      "4238\t19 * 223\tFalse\n",
      "4240\t3^3 * 157\tFalse\n",
      "4242\t4241\tTrue\n",
      "4244\t4243\tFalse\n",
      "4246\t3 * 5 * 283\tFalse\n",
      "4248\t31 * 137\tFalse\n",
      "4250\t7 * 607\tTrue\n",
      "4252\t3 * 13 * 109\tFalse\n",
      "4254\t4253\tFalse\n",
      "4256\t5 * 23 * 37\tFalse\n",
      "4258\t3^2 * 11 * 43\tFalse\n",
      "4260\t4259\tTrue\n",
      "4262\t4261\tTrue\n",
      "4264\t3 * 7^2 * 29\tFalse\n",
      "4266\t5 * 853\tFalse\n",
      "4268\t17 * 251\tFalse\n",
      "4270\t3 * 1423\tFalse\n",
      "4272\t4271\tTrue\n",
      "4274\t4273\tFalse\n",
      "4276\t3^2 * 5^2 * 19\tFalse\n",
      "4278\t7 * 13 * 47\tFalse\n",
      "4280\t11 * 389\tTrue\n",
      "4282\t3 * 1427\tFalse\n",
      "4284\t4283\tFalse\n",
      "4286\t5 * 857\tFalse\n",
      "4288\t3 * 1429\tFalse\n",
      "4290\t4289\tTrue\n",
      "4292\t7 * 613\tTrue\n",
      "4294\t3^4 * 53\tFalse\n",
      "4296\t5 * 859\tFalse\n",
      "4298\t4297\tFalse\n",
      "4300\t3 * 1433\tFalse\n",
      "4302\t11 * 17 * 23\tTrue\n",
      "4304\t13 * 331\tFalse\n",
      "4306\t3 * 5 * 7 * 41\tFalse\n",
      "4308\t59 * 73\tFalse\n",
      "4310\t31 * 139\tTrue\n",
      "4312\t3^2 * 479\tFalse\n",
      "4314\t19 * 227\tFalse\n",
      "4316\t5 * 863\tFalse\n",
      "4318\t3 * 1439\tFalse\n",
      "4320\t7 * 617\tTrue\n",
      "4322\t29 * 149\tTrue\n",
      "4324\t3 * 11 * 131\tFalse\n",
      "4326\t5^2 * 173\tFalse\n",
      "4328\t4327\tFalse\n",
      "4330\t3^2 * 13 * 37\tTrue\n",
      "4332\t61 * 71\tTrue\n",
      "4334\t7 * 619\tFalse\n",
      "4336\t3 * 5 * 17^2\tFalse\n",
      "4338\t4337\tFalse\n",
      "4340\t4339\tTrue\n",
      "4342\t3 * 1447\tFalse\n",
      "4344\t43 * 101\tFalse\n",
      "4346\t5 * 11 * 79\tFalse\n",
      "4348\t3^3 * 7 * 23\tFalse\n",
      "4350\t4349\tTrue\n",
      "4352\t19 * 229\tTrue\n",
      "4354\t3 * 1451\tFalse\n",
      "4356\t5 * 13 * 67\tFalse\n",
      "4358\t4357\tFalse\n",
      "4360\t3 * 1453\tFalse\n",
      "4362\t7^2 * 89\tTrue\n",
      "4364\t4363\tFalse\n",
      "4366\t3^2 * 5 * 97\tFalse\n",
      "4368\t11 * 397\tFalse\n",
      "4370\t17 * 257\tTrue\n",
      "4372\t3 * 31 * 47\tFalse\n",
      "4374\t4373\tFalse\n",
      "4376\t5^4 * 7\tFalse\n",
      "4378\t3 * 1459\tFalse\n",
      "4380\t29 * 151\tTrue\n",
      "4382\t13 * 337\tTrue\n",
      "4384\t3^2 * 487\tFalse\n",
      "4386\t5 * 877\tFalse\n",
      "4388\t41 * 107\tFalse\n",
      "4390\t3 * 7 * 11 * 19\tTrue\n",
      "4392\t4391\tTrue\n",
      "4394\t23 * 191\tFalse\n",
      "4396\t3 * 5 * 293\tFalse\n",
      "4398\t4397\tFalse\n",
      "4400\t53 * 83\tTrue\n",
      "4402\t3^3 * 163\tFalse\n",
      "4404\t7 * 17 * 37\tFalse\n",
      "4406\t5 * 881\tFalse\n",
      "4408\t3 * 13 * 113\tFalse\n",
      "4410\t4409\tTrue\n",
      "4412\t11 * 401\tTrue\n",
      "4414\t3 * 1471\tFalse\n",
      "4416\t5 * 883\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4418\t7 * 631\tFalse\n",
      "4420\t3^2 * 491\tFalse\n",
      "4422\t4421\tTrue\n",
      "4424\t4423\tFalse\n",
      "4426\t3 * 5^2 * 59\tFalse\n",
      "4428\t19 * 233\tFalse\n",
      "4430\t43 * 103\tTrue\n",
      "4432\t3 * 7 * 211\tFalse\n",
      "4434\t11 * 13 * 31\tFalse\n",
      "4436\t5 * 887\tFalse\n",
      "4438\t3^2 * 17 * 29\tFalse\n",
      "4440\t23 * 193\tTrue\n",
      "4442\t4441\tTrue\n",
      "4444\t3 * 1481\tFalse\n",
      "4446\t5 * 7 * 127\tFalse\n",
      "4448\t4447\tFalse\n",
      "4450\t3 * 1483\tFalse\n",
      "4452\t4451\tTrue\n",
      "4454\t61 * 73\tFalse\n",
      "4456\t3^4 * 5 * 11\tFalse\n",
      "4458\t4457\tFalse\n",
      "4460\t7^3 * 13\tTrue\n",
      "4462\t3 * 1487\tFalse\n",
      "4464\t4463\tFalse\n",
      "4466\t5 * 19 * 47\tFalse\n",
      "4468\t3 * 1489\tFalse\n",
      "4470\t41 * 109\tTrue\n",
      "4472\t17 * 263\tTrue\n",
      "4474\t3^2 * 7 * 71\tFalse\n",
      "4476\t5^2 * 179\tFalse\n",
      "4478\t11^2 * 37\tFalse\n",
      "4480\t3 * 1493\tFalse\n",
      "4482\t4481\tTrue\n",
      "4484\t4483\tFalse\n",
      "4486\t3 * 5 * 13 * 23\tFalse\n",
      "4488\t7 * 641\tFalse\n",
      "4490\t67^2\tTrue\n",
      "4492\t3^2 * 499\tFalse\n",
      "4494\t4493\tFalse\n",
      "4496\t5 * 29 * 31\tFalse\n",
      "4498\t3 * 1499\tFalse\n",
      "4500\t11 * 409\tTrue\n",
      "4502\t7 * 643\tTrue\n",
      "4504\t3 * 19 * 79\tFalse\n",
      "4506\t5 * 17 * 53\tFalse\n",
      "4508\t4507\tFalse\n",
      "4510\t3^3 * 167\tFalse\n",
      "4512\t13 * 347\tTrue\n",
      "4514\t4513\tFalse\n",
      "4516\t3 * 5 * 7 * 43\tFalse\n",
      "4518\t4517\tFalse\n",
      "4520\t4519\tTrue\n",
      "4522\t3 * 11 * 137\tTrue\n",
      "4524\t4523\tFalse\n",
      "4526\t5^2 * 181\tFalse\n",
      "4528\t3^2 * 503\tFalse\n",
      "4530\t7 * 647\tTrue\n",
      "4532\t23 * 197\tTrue\n",
      "4534\t3 * 1511\tFalse\n",
      "4536\t5 * 907\tFalse\n",
      "4538\t13 * 349\tFalse\n",
      "4540\t3 * 17 * 89\tFalse\n",
      "4542\t19 * 239\tTrue\n",
      "4544\t7 * 11 * 59\tFalse\n",
      "4546\t3^2 * 5 * 101\tFalse\n",
      "4548\t4547\tFalse\n",
      "4550\t4549\tTrue\n",
      "4552\t3 * 37 * 41\tFalse\n",
      "4554\t29 * 157\tFalse\n",
      "4556\t5 * 911\tFalse\n",
      "4558\t3 * 7^2 * 31\tFalse\n",
      "4560\t47 * 97\tTrue\n",
      "4562\t4561\tTrue\n",
      "4564\t3^3 * 13^2\tFalse\n",
      "4566\t5 * 11 * 83\tFalse\n",
      "4568\t4567\tFalse\n",
      "4570\t3 * 1523\tFalse\n",
      "4572\t7 * 653\tTrue\n",
      "4574\t17 * 269\tFalse\n",
      "4576\t3 * 5^2 * 61\tFalse\n",
      "4578\t23 * 199\tFalse\n",
      "4580\t19 * 241\tTrue\n",
      "4582\t3^2 * 509\tFalse\n",
      "4584\t4583\tFalse\n",
      "4586\t5 * 7 * 131\tFalse\n",
      "4588\t3 * 11 * 139\tFalse\n",
      "4590\t13 * 353\tTrue\n",
      "4592\t4591\tTrue\n",
      "4594\t3 * 1531\tFalse\n",
      "4596\t5 * 919\tFalse\n",
      "4598\t4597\tFalse\n",
      "4600\t3^2 * 7 * 73\tTrue\n",
      "4602\t43 * 107\tTrue\n",
      "4604\t4603\tFalse\n",
      "4606\t3 * 5 * 307\tFalse\n",
      "4608\t17 * 271\tFalse\n",
      "4610\t11 * 419\tTrue\n",
      "4612\t3 * 29 * 53\tFalse\n",
      "4614\t7 * 659\tFalse\n",
      "4616\t5 * 13 * 71\tFalse\n",
      "4618\t3^5 * 19\tFalse\n",
      "4620\t31 * 149\tTrue\n",
      "4622\t4621\tTrue\n",
      "4624\t3 * 23 * 67\tFalse\n",
      "4626\t5^3 * 37\tFalse\n",
      "4628\t7 * 661\tFalse\n",
      "4630\t3 * 1543\tFalse\n",
      "4632\t11 * 421\tTrue\n",
      "4634\t41 * 113\tFalse\n",
      "4636\t3^2 * 5 * 103\tFalse\n",
      "4638\t4637\tFalse\n",
      "4640\t4639\tTrue\n",
      "4642\t3 * 7 * 13 * 17\tTrue\n",
      "4644\t4643\tFalse\n",
      "4646\t5 * 929\tFalse\n",
      "4648\t3 * 1549\tFalse\n",
      "4650\t4649\tTrue\n",
      "4652\t4651\tTrue\n",
      "4654\t3^2 * 11 * 47\tFalse\n",
      "4656\t5 * 7^2 * 19\tFalse\n",
      "4658\t4657\tFalse\n",
      "4660\t3 * 1553\tFalse\n",
      "4662\t59 * 79\tTrue\n",
      "4664\t4663\tFalse\n",
      "4666\t3 * 5 * 311\tFalse\n",
      "4668\t13 * 359\tFalse\n",
      "4670\t7 * 23 * 29\tTrue\n",
      "4672\t3^3 * 173\tFalse\n",
      "4674\t4673\tFalse\n",
      "4676\t5^2 * 11 * 17\tFalse\n",
      "4678\t3 * 1559\tFalse\n",
      "4680\t4679\tTrue\n",
      "4682\t31 * 151\tTrue\n",
      "4684\t3 * 7 * 223\tFalse\n",
      "4686\t5 * 937\tFalse\n",
      "4688\t43 * 109\tFalse\n",
      "4690\t3^2 * 521\tFalse\n",
      "4692\t4691\tTrue\n",
      "4694\t13 * 19^2\tFalse\n",
      "4696\t3 * 5 * 313\tFalse\n",
      "4698\t7 * 11 * 61\tFalse\n",
      "4700\t37 * 127\tTrue\n",
      "4702\t3 * 1567\tFalse\n",
      "4704\t4703\tFalse\n",
      "4706\t5 * 941\tFalse\n",
      "4708\t3^2 * 523\tFalse\n",
      "4710\t17 * 277\tTrue\n",
      "4712\t7 * 673\tTrue\n",
      "4714\t3 * 1571\tFalse\n",
      "4716\t5 * 23 * 41\tFalse\n",
      "4718\t53 * 89\tFalse\n",
      "4720\t3 * 11^2 * 13\tFalse\n",
      "4722\t4721\tTrue\n",
      "4724\t4723\tFalse\n",
      "4726\t3^3 * 5^2 * 7\tFalse\n",
      "4728\t29 * 163\tFalse\n",
      "4730\t4729\tTrue\n",
      "4732\t3 * 19 * 83\tFalse\n",
      "4734\t4733\tFalse\n",
      "4736\t5 * 947\tFalse\n",
      "4738\t3 * 1579\tFalse\n",
      "4740\t7 * 677\tTrue\n",
      "4742\t11 * 431\tTrue\n",
      "4744\t3^2 * 17 * 31\tFalse\n",
      "4746\t5 * 13 * 73\tFalse\n",
      "4748\t47 * 101\tFalse\n",
      "4750\t3 * 1583\tFalse\n",
      "4752\t4751\tTrue\n",
      "4754\t7^2 * 97\tFalse\n",
      "4756\t3 * 5 * 317\tFalse\n",
      "4758\t67 * 71\tFalse\n",
      "4760\t4759\tTrue\n",
      "4762\t3^2 * 23^2\tFalse\n",
      "4764\t11 * 433\tFalse\n",
      "4766\t5 * 953\tFalse\n",
      "4768\t3 * 7 * 227\tFalse\n",
      "4770\t19 * 251\tTrue\n",
      "4772\t13 * 367\tTrue\n",
      "4774\t3 * 37 * 43\tFalse\n",
      "4776\t5^2 * 191\tFalse\n",
      "4778\t17 * 281\tFalse\n",
      "4780\t3^4 * 59\tFalse\n",
      "4782\t7 * 683\tTrue\n",
      "4784\t4783\tFalse\n",
      "4786\t3 * 5 * 11 * 29\tFalse\n",
      "4788\t4787\tFalse\n",
      "4790\t4789\tTrue\n",
      "4792\t3 * 1597\tFalse\n",
      "4794\t4793\tFalse\n",
      "4796\t5 * 7 * 137\tFalse\n",
      "4798\t3^2 * 13 * 41\tFalse\n",
      "4800\t4799\tTrue\n",
      "4802\t4801\tTrue\n",
      "4804\t3 * 1601\tFalse\n",
      "4806\t5 * 31^2\tFalse\n",
      "4808\t11 * 19 * 23\tFalse\n",
      "4810\t3 * 7 * 229\tFalse\n",
      "4812\t17 * 283\tTrue\n",
      "4814\t4813\tFalse\n",
      "4816\t3^2 * 5 * 107\tFalse\n",
      "4818\t4817\tFalse\n",
      "4820\t61 * 79\tTrue\n",
      "4822\t3 * 1607\tFalse\n",
      "4824\t7 * 13 * 53\tFalse\n",
      "4826\t5^2 * 193\tFalse\n",
      "4828\t3 * 1609\tFalse\n",
      "4830\t11 * 439\tTrue\n",
      "4832\t4831\tTrue\n",
      "4834\t3^3 * 179\tFalse\n",
      "4836\t5 * 967\tFalse\n",
      "4838\t7 * 691\tFalse\n",
      "4840\t3 * 1613\tFalse\n",
      "4842\t47 * 103\tTrue\n",
      "4844\t29 * 167\tFalse\n",
      "4846\t3 * 5 * 17 * 19\tFalse\n",
      "4848\t37 * 131\tFalse\n",
      "4850\t13 * 373\tTrue\n",
      "4852\t3^2 * 7^2 * 11\tTrue\n",
      "4854\t23 * 211\tFalse\n",
      "4856\t5 * 971\tFalse\n",
      "4858\t3 * 1619\tFalse\n",
      "4860\t43 * 113\tTrue\n",
      "4862\t4861\tTrue\n",
      "4864\t3 * 1621\tFalse\n",
      "4866\t5 * 7 * 139\tFalse\n",
      "4868\t31 * 157\tFalse\n",
      "4870\t3^2 * 541\tFalse\n",
      "4872\t4871\tTrue\n",
      "4874\t11 * 443\tFalse\n",
      "4876\t3 * 5^3 * 13\tFalse\n",
      "4878\t4877\tFalse\n",
      "4880\t7 * 17 * 41\tTrue\n",
      "4882\t3 * 1627\tFalse\n",
      "4884\t19 * 257\tFalse\n",
      "4886\t5 * 977\tFalse\n",
      "4888\t3^3 * 181\tFalse\n",
      "4890\t4889\tTrue\n",
      "4892\t67 * 73\tTrue\n",
      "4894\t3 * 7 * 233\tFalse\n",
      "4896\t5 * 11 * 89\tFalse\n",
      "4898\t59 * 83\tFalse\n",
      "4900\t3 * 23 * 71\tFalse\n",
      "4902\t13^2 * 29\tTrue\n",
      "4904\t4903\tFalse\n",
      "4906\t3^2 * 5 * 109\tFalse\n",
      "4908\t7 * 701\tFalse\n",
      "4910\t4909\tTrue\n",
      "4912\t3 * 1637\tFalse\n",
      "4914\t17^3\tFalse\n",
      "4916\t5 * 983\tFalse\n",
      "4918\t3 * 11 * 149\tFalse\n",
      "4920\t4919\tTrue\n",
      "4922\t7 * 19 * 37\tTrue\n",
      "4924\t3^2 * 547\tFalse\n",
      "4926\t5^2 * 197\tFalse\n",
      "4928\t13 * 379\tFalse\n",
      "4930\t3 * 31 * 53\tTrue\n",
      "4932\t4931\tTrue\n",
      "4934\t4933\tFalse\n",
      "4936\t3 * 5 * 7 * 47\tFalse\n",
      "4938\t4937\tFalse\n",
      "4940\t11 * 449\tTrue\n",
      "4942\t3^4 * 61\tFalse\n",
      "4944\t4943\tFalse\n",
      "4946\t5 * 23 * 43\tFalse\n",
      "4948\t3 * 17 * 97\tFalse\n",
      "4950\t7^2 * 101\tTrue\n",
      "4952\t4951\tTrue\n",
      "4954\t3 * 13 * 127\tFalse\n",
      "4956\t5 * 991\tFalse\n",
      "4958\t4957\tFalse\n",
      "4960\t3^2 * 19 * 29\tFalse\n",
      "4962\t11^2 * 41\tTrue\n",
      "4964\t7 * 709\tFalse\n",
      "4966\t3 * 5 * 331\tFalse\n",
      "4968\t4967\tFalse\n",
      "4970\t4969\tTrue\n",
      "4972\t3 * 1657\tFalse\n",
      "4974\t4973\tFalse\n",
      "4976\t5^2 * 199\tFalse\n",
      "4978\t3^2 * 7 * 79\tFalse\n",
      "4980\t13 * 383\tTrue\n",
      "4982\t17 * 293\tTrue\n",
      "4984\t3 * 11 * 151\tFalse\n",
      "4986\t5 * 997\tFalse\n",
      "4988\t4987\tFalse\n",
      "4990\t3 * 1663\tFalse\n",
      "4992\t7 * 23 * 31\tTrue\n",
      "4994\t4993\tFalse\n",
      "4996\t3^3 * 5 * 37\tFalse\n",
      "4998\t19 * 263\tFalse\n",
      "5000\t4999\tTrue\n",
      "5002\t3 * 1667\tFalse\n",
      "5004\t5003\tFalse\n",
      "5006\t5 * 7 * 11 * 13\tFalse\n",
      "5008\t3 * 1669\tFalse\n",
      "5010\t5009\tTrue\n",
      "5012\t5011\tTrue\n",
      "5014\t3^2 * 557\tFalse\n",
      "5016\t5 * 17 * 59\tFalse\n",
      "5018\t29 * 173\tFalse\n",
      "5020\t3 * 7 * 239\tFalse\n",
      "5022\t5021\tTrue\n",
      "5024\t5023\tFalse\n",
      "5026\t3 * 5^2 * 67\tFalse\n",
      "5028\t11 * 457\tFalse\n",
      "5030\t47 * 107\tTrue\n",
      "5032\t3^2 * 13 * 43\tTrue\n",
      "5034\t7 * 719\tFalse\n",
      "5036\t5 * 19 * 53\tFalse\n",
      "5038\t3 * 23 * 73\tFalse\n",
      "5040\t5039\tTrue\n",
      "5042\t71^2\tTrue\n",
      "5044\t3 * 41^2\tFalse\n",
      "5046\t5 * 1009\tFalse\n",
      "5048\t7^2 * 103\tFalse\n",
      "5050\t3^3 * 11 * 17\tFalse\n",
      "5052\t5051\tTrue\n",
      "5054\t31 * 163\tFalse\n",
      "5056\t3 * 5 * 337\tFalse\n",
      "5058\t13 * 389\tFalse\n",
      "5060\t5059\tTrue\n",
      "5062\t3 * 7 * 241\tFalse\n",
      "5064\t61 * 83\tFalse\n",
      "5066\t5 * 1013\tFalse\n",
      "5068\t3^2 * 563\tFalse\n",
      "5070\t37 * 137\tTrue\n",
      "5072\t11 * 461\tTrue\n",
      "5074\t3 * 19 * 89\tFalse\n",
      "5076\t5^2 * 7 * 29\tFalse\n",
      "5078\t5077\tFalse\n",
      "5080\t3 * 1693\tFalse\n",
      "5082\t5081\tTrue\n",
      "5084\t13 * 17 * 23\tFalse\n",
      "5086\t3^2 * 5 * 113\tFalse\n",
      "5088\t5087\tFalse\n",
      "5090\t7 * 727\tTrue\n",
      "5092\t3 * 1697\tFalse\n",
      "5094\t11 * 463\tFalse\n",
      "5096\t5 * 1019\tFalse\n",
      "5098\t3 * 1699\tFalse\n",
      "5100\t5099\tTrue\n",
      "5102\t5101\tTrue\n",
      "5104\t3^6 * 7\tFalse\n",
      "5106\t5 * 1021\tFalse\n",
      "5108\t5107\tFalse\n",
      "5110\t3 * 13 * 131\tTrue\n",
      "5112\t19 * 269\tTrue\n",
      "5114\t5113\tFalse\n",
      "5116\t3 * 5 * 11 * 31\tFalse\n",
      "5118\t7 * 17 * 43\tFalse\n",
      "5120\t5119\tTrue\n",
      "5122\t3^2 * 569\tFalse\n",
      "5124\t47 * 109\tFalse\n",
      "5126\t5^3 * 41\tFalse\n",
      "5128\t3 * 1709\tFalse\n",
      "5130\t23 * 223\tTrue\n",
      "5132\t7 * 733\tTrue\n",
      "5134\t3 * 29 * 59\tFalse\n",
      "5136\t5 * 13 * 79\tFalse\n",
      "5138\t11 * 467\tFalse\n",
      "5140\t3^2 * 571\tFalse\n",
      "5142\t53 * 97\tTrue\n",
      "5144\t37 * 139\tFalse\n",
      "5146\t3 * 5 * 7^3\tFalse\n",
      "5148\t5147\tFalse\n",
      "5150\t19 * 271\tTrue\n",
      "5152\t3 * 17 * 101\tFalse\n",
      "5154\t5153\tFalse\n",
      "5156\t5 * 1031\tFalse\n",
      "5158\t3^3 * 191\tFalse\n",
      "5160\t7 * 11 * 67\tTrue\n",
      "5162\t13 * 397\tTrue\n",
      "5164\t3 * 1721\tFalse\n",
      "5166\t5 * 1033\tFalse\n",
      "5168\t5167\tFalse\n",
      "5170\t3 * 1723\tFalse\n",
      "5172\t5171\tTrue\n",
      "5174\t7 * 739\tFalse\n",
      "5176\t3^2 * 5^2 * 23\tFalse\n",
      "5178\t31 * 167\tFalse\n",
      "5180\t5179\tTrue\n",
      "5182\t3 * 11 * 157\tTrue\n",
      "5184\t71 * 73\tFalse\n",
      "5186\t5 * 17 * 61\tFalse\n",
      "5188\t3 * 7 * 13 * 19\tFalse\n",
      "5190\t5189\tTrue\n",
      "5192\t29 * 179\tTrue\n",
      "5194\t3^2 * 577\tFalse\n",
      "5196\t5 * 1039\tFalse\n",
      "5198\t5197\tFalse\n",
      "5200\t3 * 1733\tFalse\n",
      "5202\t7 * 743\tTrue\n",
      "5204\t11^2 * 43\tFalse\n",
      "5206\t3 * 5 * 347\tFalse\n",
      "5208\t41 * 127\tFalse\n",
      "5210\t5209\tTrue\n",
      "5212\t3^3 * 193\tFalse\n",
      "5214\t13 * 401\tFalse\n",
      "5216\t5 * 7 * 149\tFalse\n",
      "5218\t3 * 37 * 47\tFalse\n",
      "5220\t17 * 307\tTrue\n",
      "5222\t23 * 227\tTrue\n",
      "5224\t3 * 1741\tFalse\n",
      "5226\t5^2 * 11 * 19\tFalse\n",
      "5228\t5227\tFalse\n",
      "5230\t3^2 * 7 * 83\tFalse\n",
      "5232\t5231\tTrue\n",
      "5234\t5233\tFalse\n",
      "5236\t3 * 5 * 349\tFalse\n",
      "5238\t5237\tFalse\n",
      "5240\t13^2 * 31\tTrue\n",
      "5242\t3 * 1747\tFalse\n",
      "5244\t7^2 * 107\tFalse\n",
      "5246\t5 * 1049\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5248\t3^2 * 11 * 53\tFalse\n",
      "5250\t29 * 181\tTrue\n",
      "5252\t59 * 89\tTrue\n",
      "5254\t3 * 17 * 103\tFalse\n",
      "5256\t5 * 1051\tFalse\n",
      "5258\t7 * 751\tFalse\n",
      "5260\t3 * 1753\tFalse\n",
      "5262\t5261\tTrue\n",
      "5264\t19 * 277\tFalse\n",
      "5266\t3^4 * 5 * 13\tFalse\n",
      "5268\t23 * 229\tFalse\n",
      "5270\t11 * 479\tTrue\n",
      "5272\t3 * 7 * 251\tFalse\n",
      "5274\t5273\tFalse\n",
      "5276\t5^2 * 211\tFalse\n",
      "5278\t3 * 1759\tFalse\n",
      "5280\t5279\tTrue\n",
      "5282\t5281\tTrue\n",
      "5284\t3^2 * 587\tFalse\n",
      "5286\t5 * 7 * 151\tFalse\n",
      "5288\t17 * 311\tFalse\n",
      "5290\t3 * 41 * 43\tFalse\n",
      "5292\t11 * 13 * 37\tTrue\n",
      "5294\t67 * 79\tFalse\n",
      "5296\t3 * 5 * 353\tFalse\n",
      "5298\t5297\tFalse\n",
      "5300\t7 * 757\tTrue\n",
      "5302\t3^2 * 19 * 31\tTrue\n",
      "5304\t5303\tFalse\n",
      "5306\t5 * 1061\tFalse\n",
      "5308\t3 * 29 * 61\tFalse\n",
      "5310\t5309\tTrue\n",
      "5312\t47 * 113\tTrue\n",
      "5314\t3 * 7 * 11 * 23\tFalse\n",
      "5316\t5 * 1063\tFalse\n",
      "5318\t13 * 409\tFalse\n",
      "5320\t3^3 * 197\tFalse\n",
      "5322\t17 * 313\tTrue\n",
      "5324\t5323\tFalse\n",
      "5326\t3 * 5^2 * 71\tFalse\n",
      "5328\t7 * 761\tFalse\n",
      "5330\t73^2\tTrue\n",
      "5332\t3 * 1777\tFalse\n",
      "5334\t5333\tFalse\n",
      "5336\t5 * 11 * 97\tFalse\n",
      "5338\t3^2 * 593\tFalse\n",
      "5340\t19 * 281\tTrue\n",
      "5342\t7^2 * 109\tTrue\n",
      "5344\t3 * 13 * 137\tFalse\n",
      "5346\t5 * 1069\tFalse\n",
      "5348\t5347\tFalse\n",
      "5350\t3 * 1783\tFalse\n",
      "5352\t5351\tTrue\n",
      "5354\t53 * 101\tFalse\n",
      "5356\t3^2 * 5 * 7 * 17\tFalse\n",
      "5358\t11 * 487\tFalse\n",
      "5360\t23 * 233\tTrue\n",
      "5362\t3 * 1787\tFalse\n",
      "5364\t31 * 173\tFalse\n",
      "5366\t5 * 29 * 37\tFalse\n",
      "5368\t3 * 1789\tFalse\n",
      "5370\t7 * 13 * 59\tTrue\n",
      "5372\t41 * 131\tTrue\n",
      "5374\t3^3 * 199\tFalse\n",
      "5376\t5^3 * 43\tFalse\n",
      "5378\t19 * 283\tFalse\n",
      "5380\t3 * 11 * 163\tTrue\n",
      "5382\t5381\tTrue\n",
      "5384\t7 * 769\tFalse\n",
      "5386\t3 * 5 * 359\tFalse\n",
      "5388\t5387\tFalse\n",
      "5390\t17 * 317\tTrue\n",
      "5392\t3^2 * 599\tFalse\n",
      "5394\t5393\tFalse\n",
      "5396\t5 * 13 * 83\tFalse\n",
      "5398\t3 * 7 * 257\tFalse\n",
      "5400\t5399\tTrue\n",
      "5402\t11 * 491\tTrue\n",
      "5404\t3 * 1801\tFalse\n",
      "5406\t5 * 23 * 47\tFalse\n",
      "5408\t5407\tFalse\n",
      "5410\t3^2 * 601\tFalse\n",
      "5412\t7 * 773\tTrue\n",
      "5414\t5413\tFalse\n",
      "5416\t3 * 5 * 19^2\tFalse\n",
      "5418\t5417\tFalse\n",
      "5420\t5419\tTrue\n",
      "5422\t3 * 13 * 139\tTrue\n",
      "5424\t11 * 17 * 29\tFalse\n",
      "5426\t5^2 * 7 * 31\tFalse\n",
      "5428\t3^4 * 67\tFalse\n",
      "5430\t61 * 89\tTrue\n",
      "5432\t5431\tTrue\n",
      "5434\t3 * 1811\tFalse\n",
      "5436\t5 * 1087\tFalse\n",
      "5438\t5437\tFalse\n",
      "5440\t3 * 7^2 * 37\tFalse\n",
      "5442\t5441\tTrue\n",
      "5444\t5443\tFalse\n",
      "5446\t3^2 * 5 * 11^2\tFalse\n",
      "5448\t13 * 419\tFalse\n",
      "5450\t5449\tTrue\n",
      "5452\t3 * 23 * 79\tFalse\n",
      "5454\t7 * 19 * 41\tFalse\n",
      "5456\t5 * 1091\tFalse\n",
      "5458\t3 * 17 * 107\tFalse\n",
      "5460\t53 * 103\tTrue\n",
      "5462\t43 * 127\tTrue\n",
      "5464\t3^2 * 607\tFalse\n",
      "5466\t5 * 1093\tFalse\n",
      "5468\t7 * 11 * 71\tFalse\n",
      "5470\t3 * 1823\tFalse\n",
      "5472\t5471\tTrue\n",
      "5474\t13 * 421\tFalse\n",
      "5476\t3 * 5^2 * 73\tFalse\n",
      "5478\t5477\tFalse\n",
      "5480\t5479\tTrue\n",
      "5482\t3^3 * 7 * 29\tFalse\n",
      "5484\t5483\tFalse\n",
      "5486\t5 * 1097\tFalse\n",
      "5488\t3 * 31 * 59\tFalse\n",
      "5490\t11 * 499\tTrue\n",
      "5492\t17^2 * 19\tTrue\n",
      "5494\t3 * 1831\tFalse\n",
      "5496\t5 * 7 * 157\tFalse\n",
      "5498\t23 * 239\tFalse\n",
      "5500\t3^2 * 13 * 47\tFalse\n",
      "5502\t5501\tTrue\n",
      "5504\t5503\tFalse\n",
      "5506\t3 * 5 * 367\tFalse\n",
      "5508\t5507\tFalse\n",
      "5510\t7 * 787\tTrue\n",
      "5512\t3 * 11 * 167\tTrue\n",
      "5514\t37 * 149\tFalse\n",
      "5516\t5 * 1103\tFalse\n",
      "5518\t3^2 * 613\tFalse\n",
      "5520\t5519\tTrue\n",
      "5522\t5521\tTrue\n",
      "5524\t3 * 7 * 263\tFalse\n",
      "5526\t5^2 * 13 * 17\tFalse\n",
      "5528\t5527\tFalse\n",
      "5530\t3 * 19 * 97\tTrue\n",
      "5532\t5531\tTrue\n",
      "5534\t11 * 503\tFalse\n",
      "5536\t3^3 * 5 * 41\tFalse\n",
      "5538\t7^2 * 113\tFalse\n",
      "5540\t29 * 191\tTrue\n",
      "5542\t3 * 1847\tFalse\n",
      "5544\t23 * 241\tFalse\n",
      "5546\t5 * 1109\tFalse\n",
      "5548\t3 * 43^2\tFalse\n",
      "5550\t31 * 179\tTrue\n",
      "5552\t7 * 13 * 61\tTrue\n",
      "5554\t3^2 * 617\tFalse\n",
      "5556\t5 * 11 * 101\tFalse\n",
      "5558\t5557\tFalse\n",
      "5560\t3 * 17 * 109\tFalse\n",
      "5562\t67 * 83\tTrue\n",
      "5564\t5563\tFalse\n",
      "5566\t3 * 5 * 7 * 53\tFalse\n",
      "5568\t19 * 293\tFalse\n",
      "5570\t5569\tTrue\n",
      "5572\t3^2 * 619\tFalse\n",
      "5574\t5573\tFalse\n",
      "5576\t5^2 * 223\tFalse\n",
      "5578\t3 * 11 * 13^2\tFalse\n",
      "5580\t7 * 797\tTrue\n",
      "5582\t5581\tTrue\n",
      "5584\t3 * 1861\tFalse\n",
      "5586\t5 * 1117\tFalse\n",
      "5588\t37 * 151\tFalse\n",
      "5590\t3^5 * 23\tFalse\n",
      "5592\t5591\tTrue\n",
      "5594\t7 * 17 * 47\tFalse\n",
      "5596\t3 * 5 * 373\tFalse\n",
      "5598\t29 * 193\tFalse\n",
      "5600\t11 * 509\tTrue\n",
      "5602\t3 * 1867\tFalse\n",
      "5604\t13 * 431\tFalse\n",
      "5606\t5 * 19 * 59\tFalse\n",
      "5608\t3^2 * 7 * 89\tFalse\n",
      "5610\t71 * 79\tTrue\n",
      "5612\t31 * 181\tTrue\n",
      "5614\t3 * 1871\tFalse\n",
      "5616\t5 * 1123\tFalse\n",
      "5618\t41 * 137\tFalse\n",
      "5620\t3 * 1873\tFalse\n",
      "5622\t7 * 11 * 73\tTrue\n",
      "5624\t5623\tFalse\n",
      "5626\t3^2 * 5^4\tFalse\n",
      "5628\t17 * 331\tFalse\n",
      "5630\t13 * 433\tTrue\n",
      "5632\t3 * 1877\tFalse\n",
      "5634\t43 * 131\tFalse\n",
      "5636\t5 * 7^2 * 23\tFalse\n",
      "5638\t3 * 1879\tFalse\n",
      "5640\t5639\tTrue\n",
      "5642\t5641\tTrue\n",
      "5644\t3^3 * 11 * 19\tFalse\n",
      "5646\t5 * 1129\tFalse\n",
      "5648\t5647\tFalse\n",
      "5650\t3 * 7 * 269\tFalse\n",
      "5652\t5651\tTrue\n",
      "5654\t5653\tFalse\n",
      "5656\t3 * 5 * 13 * 29\tFalse\n",
      "5658\t5657\tFalse\n",
      "5660\t5659\tTrue\n",
      "5662\t3^2 * 17 * 37\tTrue\n",
      "5664\t7 * 809\tFalse\n",
      "5666\t5 * 11 * 103\tFalse\n",
      "5668\t3 * 1889\tFalse\n",
      "5670\t5669\tTrue\n",
      "5672\t53 * 107\tTrue\n",
      "5674\t3 * 31 * 61\tFalse\n",
      "5676\t5^2 * 227\tFalse\n",
      "5678\t7 * 811\tFalse\n",
      "5680\t3^2 * 631\tFalse\n",
      "5682\t13 * 19 * 23\tTrue\n",
      "5684\t5683\tFalse\n",
      "5686\t3 * 5 * 379\tFalse\n",
      "5688\t11^2 * 47\tFalse\n",
      "5690\t5689\tTrue\n",
      "5692\t3 * 7 * 271\tFalse\n",
      "5694\t5693\tFalse\n",
      "5696\t5 * 17 * 67\tFalse\n",
      "5698\t3^3 * 211\tFalse\n",
      "5700\t41 * 139\tTrue\n",
      "5702\t5701\tTrue\n",
      "5704\t3 * 1901\tFalse\n",
      "5706\t5 * 7 * 163\tFalse\n",
      "5708\t13 * 439\tFalse\n",
      "5710\t3 * 11 * 173\tTrue\n",
      "5712\t5711\tTrue\n",
      "5714\t29 * 197\tFalse\n",
      "5716\t3^2 * 5 * 127\tFalse\n",
      "5718\t5717\tFalse\n",
      "5720\t7 * 19 * 43\tTrue\n",
      "5722\t3 * 1907\tFalse\n",
      "5724\t59 * 97\tFalse\n",
      "5726\t5^2 * 229\tFalse\n",
      "5728\t3 * 23 * 83\tFalse\n",
      "5730\t17 * 337\tTrue\n",
      "5732\t11 * 521\tTrue\n",
      "5734\t3^2 * 7^2 * 13\tFalse\n",
      "5736\t5 * 31 * 37\tFalse\n",
      "5738\t5737\tFalse\n",
      "5740\t3 * 1913\tFalse\n",
      "5742\t5741\tTrue\n",
      "5744\t5743\tFalse\n",
      "5746\t3 * 5 * 383\tFalse\n",
      "5748\t7 * 821\tFalse\n",
      "5750\t5749\tTrue\n",
      "5752\t3^4 * 71\tFalse\n",
      "5754\t11 * 523\tFalse\n",
      "5756\t5 * 1151\tFalse\n",
      "5758\t3 * 19 * 101\tFalse\n",
      "5760\t13 * 443\tTrue\n",
      "5762\t7 * 823\tTrue\n",
      "5764\t3 * 17 * 113\tFalse\n",
      "5766\t5 * 1153\tFalse\n",
      "5768\t73 * 79\tFalse\n",
      "5770\t3^2 * 641\tFalse\n",
      "5772\t29 * 199\tTrue\n",
      "5774\t23 * 251\tFalse\n",
      "5776\t3 * 5^2 * 7 * 11\tFalse\n",
      "5778\t53 * 109\tFalse\n",
      "5780\t5779\tTrue\n",
      "5782\t3 * 41 * 47\tFalse\n",
      "5784\t5783\tFalse\n",
      "5786\t5 * 13 * 89\tFalse\n",
      "5788\t3^2 * 643\tFalse\n",
      "5790\t7 * 827\tTrue\n",
      "5792\t5791\tTrue\n",
      "5794\t3 * 1931\tFalse\n",
      "5796\t5 * 19 * 61\tFalse\n",
      "5798\t11 * 17 * 31\tFalse\n",
      "5800\t3 * 1933\tFalse\n",
      "5802\t5801\tTrue\n",
      "5804\t7 * 829\tFalse\n",
      "5806\t3^3 * 5 * 43\tFalse\n",
      "5808\t5807\tFalse\n",
      "5810\t37 * 157\tTrue\n",
      "5812\t3 * 13 * 149\tTrue\n",
      "5814\t5813\tFalse\n",
      "5816\t5 * 1163\tFalse\n",
      "5818\t3 * 7 * 277\tFalse\n",
      "5820\t11 * 23^2\tTrue\n",
      "5822\t5821\tTrue\n",
      "5824\t3^2 * 647\tFalse\n",
      "5826\t5^2 * 233\tFalse\n",
      "5828\t5827\tFalse\n",
      "5830\t3 * 29 * 67\tFalse\n",
      "5832\t7^3 * 17\tTrue\n",
      "5834\t19 * 307\tFalse\n",
      "5836\t3 * 5 * 389\tFalse\n",
      "5838\t13 * 449\tFalse\n",
      "5840\t5839\tTrue\n",
      "5842\t3^2 * 11 * 59\tTrue\n",
      "5844\t5843\tFalse\n",
      "5846\t5 * 7 * 167\tFalse\n",
      "5848\t3 * 1949\tFalse\n",
      "5850\t5849\tTrue\n",
      "5852\t5851\tTrue\n",
      "5854\t3 * 1951\tFalse\n",
      "5856\t5 * 1171\tFalse\n",
      "5858\t5857\tFalse\n",
      "5860\t3^3 * 7 * 31\tFalse\n",
      "5862\t5861\tTrue\n",
      "5864\t11 * 13 * 41\tFalse\n",
      "5866\t3 * 5 * 17 * 23\tFalse\n",
      "5868\t5867\tFalse\n",
      "5870\t5869\tTrue\n",
      "5872\t3 * 19 * 103\tFalse\n",
      "5874\t7 * 839\tFalse\n",
      "5876\t5^3 * 47\tFalse\n",
      "5878\t3^2 * 653\tFalse\n",
      "5880\t5879\tTrue\n",
      "5882\t5881\tTrue\n",
      "5884\t3 * 37 * 53\tFalse\n",
      "5886\t5 * 11 * 107\tFalse\n",
      "5888\t7 * 29^2\tFalse\n",
      "5890\t3 * 13 * 151\tTrue\n",
      "5892\t43 * 137\tTrue\n",
      "5894\t71 * 83\tFalse\n",
      "5896\t3^2 * 5 * 131\tFalse\n",
      "5898\t5897\tFalse\n",
      "5900\t17 * 347\tTrue\n",
      "5902\t3 * 7 * 281\tFalse\n",
      "5904\t5903\tFalse\n",
      "5906\t5 * 1181\tFalse\n",
      "5908\t3 * 11 * 179\tFalse\n",
      "5910\t19 * 311\tTrue\n",
      "5912\t23 * 257\tTrue\n",
      "5914\t3^4 * 73\tFalse\n",
      "5916\t5 * 7 * 13^2\tFalse\n",
      "5918\t61 * 97\tFalse\n",
      "5920\t3 * 1973\tFalse\n",
      "5922\t31 * 191\tTrue\n",
      "5924\t5923\tFalse\n",
      "5926\t3 * 5^2 * 79\tFalse\n",
      "5928\t5927\tFalse\n",
      "5930\t7^2 * 11^2\tTrue\n",
      "5932\t3^2 * 659\tFalse\n",
      "5934\t17 * 349\tFalse\n",
      "5936\t5 * 1187\tFalse\n",
      "5938\t3 * 1979\tFalse\n",
      "5940\t5939\tTrue\n",
      "5942\t13 * 457\tTrue\n",
      "5944\t3 * 7 * 283\tFalse\n",
      "5946\t5 * 29 * 41\tFalse\n",
      "5948\t19 * 313\tFalse\n",
      "5950\t3^2 * 661\tFalse\n",
      "5952\t11 * 541\tTrue\n",
      "5954\t5953\tFalse\n",
      "5956\t3 * 5 * 397\tFalse\n",
      "5958\t7 * 23 * 37\tFalse\n",
      "5960\t59 * 101\tTrue\n",
      "5962\t3 * 1987\tFalse\n",
      "5964\t67 * 89\tFalse\n",
      "5966\t5 * 1193\tFalse\n",
      "5968\t3^3 * 13 * 17\tFalse\n",
      "5970\t47 * 127\tTrue\n",
      "5972\t7 * 853\tTrue\n",
      "5974\t3 * 11 * 181\tFalse\n",
      "5976\t5^2 * 239\tFalse\n",
      "5978\t43 * 139\tFalse\n",
      "5980\t3 * 1993\tFalse\n",
      "5982\t5981\tTrue\n",
      "5984\t31 * 193\tFalse\n",
      "5986\t3^2 * 5 * 7 * 19\tFalse\n",
      "5988\t5987\tFalse\n",
      "5990\t53 * 113\tTrue\n",
      "5992\t3 * 1997\tFalse\n",
      "5994\t13 * 461\tFalse\n",
      "5996\t5 * 11 * 109\tFalse\n",
      "5998\t3 * 1999\tFalse\n",
      "6000\t7 * 857\tTrue\n",
      "6002\t17 * 353\tTrue\n",
      "6004\t3^2 * 23 * 29\tFalse\n",
      "6006\t5 * 1201\tFalse\n",
      "6008\t6007\tFalse\n",
      "6010\t3 * 2003\tFalse\n",
      "6012\t6011\tTrue\n",
      "6014\t7 * 859\tFalse\n",
      "6016\t3 * 5 * 401\tFalse\n",
      "6018\t11 * 547\tFalse\n",
      "6020\t13 * 463\tTrue\n",
      "6022\t3^3 * 223\tFalse\n",
      "6024\t19 * 317\tFalse\n",
      "6026\t5^2 * 241\tFalse\n",
      "6028\t3 * 7^2 * 41\tFalse\n",
      "6030\t6029\tTrue\n",
      "6032\t37 * 163\tTrue\n",
      "6034\t3 * 2011\tFalse\n",
      "6036\t5 * 17 * 71\tFalse\n",
      "6038\t6037\tFalse\n",
      "6040\t3^2 * 11 * 61\tTrue\n",
      "6042\t7 * 863\tTrue\n",
      "6044\t6043\tFalse\n",
      "6046\t3 * 5 * 13 * 31\tFalse\n",
      "6048\t6047\tFalse\n",
      "6050\t23 * 263\tTrue\n",
      "6052\t3 * 2017\tFalse\n",
      "6054\t6053\tFalse\n",
      "6056\t5 * 7 * 173\tFalse\n",
      "6058\t3^2 * 673\tFalse\n",
      "6060\t73 * 83\tTrue\n",
      "6062\t11 * 19 * 29\tTrue\n",
      "6064\t3 * 43 * 47\tFalse\n",
      "6066\t5 * 1213\tFalse\n",
      "6068\t6067\tFalse\n",
      "6070\t3 * 7 * 17^2\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6072\t13 * 467\tTrue\n",
      "6074\t6073\tFalse\n",
      "6076\t3^5 * 5^2\tFalse\n",
      "6078\t59 * 103\tFalse\n",
      "6080\t6079\tTrue\n",
      "6082\t3 * 2027\tFalse\n",
      "6084\t7 * 11 * 79\tFalse\n",
      "6086\t5 * 1217\tFalse\n",
      "6088\t3 * 2029\tFalse\n",
      "6090\t6089\tTrue\n",
      "6092\t6091\tTrue\n",
      "6094\t3^2 * 677\tFalse\n",
      "6096\t5 * 23 * 53\tFalse\n",
      "6098\t7 * 13 * 67\tFalse\n",
      "6100\t3 * 19 * 107\tFalse\n",
      "6102\t6101\tTrue\n",
      "6104\t17 * 359\tFalse\n",
      "6106\t3 * 5 * 11 * 37\tFalse\n",
      "6108\t31 * 197\tFalse\n",
      "6110\t41 * 149\tTrue\n",
      "6112\t3^2 * 7 * 97\tFalse\n",
      "6114\t6113\tFalse\n",
      "6116\t5 * 1223\tFalse\n",
      "6118\t3 * 2039\tFalse\n",
      "6120\t29 * 211\tTrue\n",
      "6122\t6121\tTrue\n",
      "6124\t3 * 13 * 157\tFalse\n",
      "6126\t5^3 * 7^2\tFalse\n",
      "6128\t11 * 557\tFalse\n",
      "6130\t3^3 * 227\tFalse\n",
      "6132\t6131\tTrue\n",
      "6134\t6133\tFalse\n",
      "6136\t3 * 5 * 409\tFalse\n",
      "6138\t17 * 19^2\tFalse\n",
      "6140\t7 * 877\tTrue\n",
      "6142\t3 * 23 * 89\tFalse\n",
      "6144\t6143\tFalse\n",
      "6146\t5 * 1229\tFalse\n",
      "6148\t3^2 * 683\tFalse\n",
      "6150\t11 * 13 * 43\tTrue\n",
      "6152\t6151\tTrue\n",
      "6154\t3 * 7 * 293\tFalse\n",
      "6156\t5 * 1231\tFalse\n",
      "6158\t47 * 131\tFalse\n",
      "6160\t3 * 2053\tFalse\n",
      "6162\t61 * 101\tTrue\n",
      "6164\t6163\tFalse\n",
      "6166\t3^2 * 5 * 137\tFalse\n",
      "6168\t7 * 881\tFalse\n",
      "6170\t31 * 199\tTrue\n",
      "6172\t3 * 11^2 * 17\tFalse\n",
      "6174\t6173\tFalse\n",
      "6176\t5^2 * 13 * 19\tFalse\n",
      "6178\t3 * 29 * 71\tFalse\n",
      "6180\t37 * 167\tTrue\n",
      "6182\t7 * 883\tTrue\n",
      "6184\t3^3 * 229\tFalse\n",
      "6186\t5 * 1237\tFalse\n",
      "6188\t23 * 269\tFalse\n",
      "6190\t3 * 2063\tFalse\n",
      "6192\t41 * 151\tTrue\n",
      "6194\t11 * 563\tFalse\n",
      "6196\t3 * 5 * 7 * 59\tFalse\n",
      "6198\t6197\tFalse\n",
      "6200\t6199\tTrue\n",
      "6202\t3^2 * 13 * 53\tTrue\n",
      "6204\t6203\tFalse\n",
      "6206\t5 * 17 * 73\tFalse\n",
      "6208\t3 * 2069\tFalse\n",
      "6210\t7 * 887\tTrue\n",
      "6212\t6211\tTrue\n",
      "6214\t3 * 19 * 109\tFalse\n",
      "6216\t5 * 11 * 113\tFalse\n",
      "6218\t6217\tFalse\n",
      "6220\t3^2 * 691\tFalse\n",
      "6222\t6221\tTrue\n",
      "6224\t7^2 * 127\tFalse\n",
      "6226\t3 * 5^2 * 83\tFalse\n",
      "6228\t13 * 479\tFalse\n",
      "6230\t6229\tTrue\n",
      "6232\t3 * 31 * 67\tFalse\n",
      "6234\t23 * 271\tFalse\n",
      "6236\t5 * 29 * 43\tFalse\n",
      "6238\t3^4 * 7 * 11\tFalse\n",
      "6240\t17 * 367\tTrue\n",
      "6242\t79^2\tTrue\n",
      "6244\t3 * 2081\tFalse\n",
      "6246\t5 * 1249\tFalse\n",
      "6248\t6247\tFalse\n",
      "6250\t3 * 2083\tFalse\n",
      "6252\t7 * 19 * 47\tTrue\n",
      "6254\t13^2 * 37\tFalse\n",
      "6256\t3^2 * 5 * 139\tFalse\n",
      "6258\t6257\tFalse\n",
      "6260\t11 * 569\tTrue\n",
      "6262\t3 * 2087\tFalse\n",
      "6264\t6263\tFalse\n",
      "6266\t5 * 7 * 179\tFalse\n",
      "6268\t3 * 2089\tFalse\n",
      "6270\t6269\tTrue\n",
      "6272\t6271\tTrue\n",
      "6274\t3^2 * 17 * 41\tFalse\n",
      "6276\t5^2 * 251\tFalse\n",
      "6278\t6277\tFalse\n",
      "6280\t3 * 7 * 13 * 23\tFalse\n",
      "6282\t11 * 571\tTrue\n",
      "6284\t61 * 103\tFalse\n",
      "6286\t3 * 5 * 419\tFalse\n",
      "6288\t6287\tFalse\n",
      "6290\t19 * 331\tTrue\n",
      "6292\t3^3 * 233\tFalse\n",
      "6294\t7 * 29 * 31\tFalse\n",
      "6296\t5 * 1259\tFalse\n",
      "6298\t3 * 2099\tFalse\n",
      "6300\t6299\tTrue\n",
      "6302\t6301\tTrue\n",
      "6304\t3 * 11 * 191\tFalse\n",
      "6306\t5 * 13 * 97\tFalse\n",
      "6308\t7 * 17 * 53\tFalse\n",
      "6310\t3^2 * 701\tFalse\n",
      "6312\t6311\tTrue\n",
      "6314\t59 * 107\tFalse\n",
      "6316\t3 * 5 * 421\tFalse\n",
      "6318\t6317\tFalse\n",
      "6320\t71 * 89\tTrue\n",
      "6322\t3 * 7^2 * 43\tFalse\n",
      "6324\t6323\tFalse\n",
      "6326\t5^2 * 11 * 23\tFalse\n",
      "6328\t3^2 * 19 * 37\tFalse\n",
      "6330\t6329\tTrue\n",
      "6332\t13 * 487\tTrue\n",
      "6334\t3 * 2111\tFalse\n",
      "6336\t5 * 7 * 181\tFalse\n",
      "6338\t6337\tFalse\n",
      "6340\t3 * 2113\tFalse\n",
      "6342\t17 * 373\tTrue\n",
      "6344\t6343\tFalse\n",
      "6346\t3^3 * 5 * 47\tFalse\n",
      "6348\t11 * 577\tFalse\n",
      "6350\t7 * 907\tTrue\n",
      "6352\t3 * 29 * 73\tFalse\n",
      "6354\t6353\tFalse\n",
      "6356\t5 * 31 * 41\tFalse\n",
      "6358\t3 * 13 * 163\tFalse\n",
      "6360\t6359\tTrue\n",
      "6362\t6361\tTrue\n",
      "6364\t3^2 * 7 * 101\tFalse\n",
      "6366\t5 * 19 * 67\tFalse\n",
      "6368\t6367\tFalse\n",
      "6370\t3 * 11 * 193\tTrue\n",
      "6372\t23 * 277\tTrue\n",
      "6374\t6373\tFalse\n",
      "6376\t3 * 5^3 * 17\tFalse\n",
      "6378\t7 * 911\tFalse\n",
      "6380\t6379\tTrue\n",
      "6382\t3^2 * 709\tFalse\n",
      "6384\t13 * 491\tFalse\n",
      "6386\t5 * 1277\tFalse\n",
      "6388\t3 * 2129\tFalse\n",
      "6390\t6389\tTrue\n",
      "6392\t7 * 11 * 83\tTrue\n",
      "6394\t3 * 2131\tFalse\n",
      "6396\t5 * 1279\tFalse\n",
      "6398\t6397\tFalse\n",
      "6400\t3^4 * 79\tFalse\n",
      "6402\t37 * 173\tTrue\n",
      "6404\t19 * 337\tFalse\n",
      "6406\t3 * 5 * 7 * 61\tFalse\n",
      "6408\t43 * 149\tFalse\n",
      "6410\t13 * 17 * 29\tTrue\n",
      "6412\t3 * 2137\tFalse\n",
      "6414\t11^2 * 53\tFalse\n",
      "6416\t5 * 1283\tFalse\n",
      "6418\t3^2 * 23 * 31\tFalse\n",
      "6420\t7^2 * 131\tTrue\n",
      "6422\t6421\tTrue\n",
      "6424\t3 * 2141\tFalse\n",
      "6426\t5^2 * 257\tFalse\n",
      "6428\t6427\tFalse\n",
      "6430\t3 * 2143\tFalse\n",
      "6432\t59 * 109\tTrue\n",
      "6434\t7 * 919\tFalse\n",
      "6436\t3^2 * 5 * 11 * 13\tFalse\n",
      "6438\t41 * 157\tFalse\n",
      "6440\t47 * 137\tTrue\n",
      "6442\t3 * 19 * 113\tTrue\n",
      "6444\t17 * 379\tFalse\n",
      "6446\t5 * 1289\tFalse\n",
      "6448\t3 * 7 * 307\tFalse\n",
      "6450\t6449\tTrue\n",
      "6452\t6451\tTrue\n",
      "6454\t3^3 * 239\tFalse\n",
      "6456\t5 * 1291\tFalse\n",
      "6458\t11 * 587\tFalse\n",
      "6460\t3 * 2153\tFalse\n",
      "6462\t7 * 13 * 71\tTrue\n",
      "6464\t23 * 281\tFalse\n",
      "6466\t3 * 5 * 431\tFalse\n",
      "6468\t29 * 223\tFalse\n",
      "6470\t6469\tTrue\n",
      "6472\t3^2 * 719\tFalse\n",
      "6474\t6473\tFalse\n",
      "6476\t5^2 * 7 * 37\tFalse\n",
      "6478\t3 * 17 * 127\tFalse\n",
      "6480\t11 * 19 * 31\tTrue\n",
      "6482\t6481\tTrue\n",
      "6484\t3 * 2161\tFalse\n",
      "6486\t5 * 1297\tFalse\n",
      "6488\t13 * 499\tFalse\n",
      "6490\t3^2 * 7 * 103\tFalse\n",
      "6492\t6491\tTrue\n",
      "6494\t43 * 151\tFalse\n",
      "6496\t3 * 5 * 433\tFalse\n",
      "6498\t73 * 89\tFalse\n",
      "6500\t67 * 97\tTrue\n",
      "6502\t3 * 11 * 197\tTrue\n",
      "6504\t7 * 929\tFalse\n",
      "6506\t5 * 1301\tFalse\n",
      "6508\t3^3 * 241\tFalse\n",
      "6510\t23 * 283\tTrue\n",
      "6512\t17 * 383\tTrue\n",
      "6514\t3 * 13 * 167\tFalse\n",
      "6516\t5 * 1303\tFalse\n",
      "6518\t7^3 * 19\tFalse\n",
      "6520\t3 * 41 * 53\tFalse\n",
      "6522\t6521\tTrue\n",
      "6524\t11 * 593\tFalse\n",
      "6526\t3^2 * 5^2 * 29\tFalse\n",
      "6528\t61 * 107\tFalse\n",
      "6530\t6529\tTrue\n",
      "6532\t3 * 7 * 311\tFalse\n",
      "6534\t47 * 139\tFalse\n",
      "6536\t5 * 1307\tFalse\n",
      "6538\t3 * 2179\tFalse\n",
      "6540\t13 * 503\tTrue\n",
      "6542\t31 * 211\tTrue\n",
      "6544\t3^2 * 727\tFalse\n",
      "6546\t5 * 7 * 11 * 17\tFalse\n",
      "6548\t6547\tFalse\n",
      "6550\t3 * 37 * 59\tTrue\n",
      "6552\t6551\tTrue\n",
      "6554\t6553\tFalse\n",
      "6556\t3 * 5 * 19 * 23\tFalse\n",
      "6558\t79 * 83\tFalse\n",
      "6560\t7 * 937\tTrue\n",
      "6562\t3^8\tFalse\n",
      "6564\t6563\tFalse\n",
      "6566\t5 * 13 * 101\tFalse\n",
      "6568\t3 * 11 * 199\tFalse\n",
      "6570\t6569\tTrue\n",
      "6572\t6571\tTrue\n",
      "6574\t3 * 7 * 313\tFalse\n",
      "6576\t5^2 * 263\tFalse\n",
      "6578\t6577\tFalse\n",
      "6580\t3^2 * 17 * 43\tTrue\n",
      "6582\t6581\tTrue\n",
      "6584\t29 * 227\tFalse\n",
      "6586\t3 * 5 * 439\tFalse\n",
      "6588\t7 * 941\tFalse\n",
      "6590\t11 * 599\tTrue\n",
      "6592\t3 * 13^3\tFalse\n",
      "6594\t19 * 347\tFalse\n",
      "6596\t5 * 1319\tFalse\n",
      "6598\t3^2 * 733\tFalse\n",
      "6600\t6599\tTrue\n",
      "6602\t7 * 23 * 41\tTrue\n",
      "6604\t3 * 31 * 71\tFalse\n",
      "6606\t5 * 1321\tFalse\n",
      "6608\t6607\tFalse\n",
      "6610\t3 * 2203\tFalse\n",
      "6612\t11 * 601\tTrue\n",
      "6614\t17 * 389\tFalse\n",
      "6616\t3^3 * 5 * 7^2\tFalse\n",
      "6618\t13 * 509\tFalse\n",
      "6620\t6619\tTrue\n",
      "6622\t3 * 2207\tFalse\n",
      "6624\t37 * 179\tFalse\n",
      "6626\t5^3 * 53\tFalse\n",
      "6628\t3 * 47^2\tFalse\n",
      "6630\t7 * 947\tTrue\n",
      "6632\t19 * 349\tTrue\n",
      "6634\t3^2 * 11 * 67\tFalse\n",
      "6636\t5 * 1327\tFalse\n",
      "6638\t6637\tFalse\n",
      "6640\t3 * 2213\tFalse\n",
      "6642\t29 * 229\tTrue\n",
      "6644\t7 * 13 * 73\tFalse\n",
      "6646\t3 * 5 * 443\tFalse\n",
      "6648\t17^2 * 23\tFalse\n",
      "6650\t61 * 109\tTrue\n",
      "6652\t3^2 * 739\tFalse\n",
      "6654\t6653\tFalse\n",
      "6656\t5 * 11^3\tFalse\n",
      "6658\t3 * 7 * 317\tFalse\n",
      "6660\t6659\tTrue\n",
      "6662\t6661\tTrue\n",
      "6664\t3 * 2221\tFalse\n",
      "6666\t5 * 31 * 43\tFalse\n",
      "6668\t59 * 113\tFalse\n",
      "6670\t3^3 * 13 * 19\tFalse\n",
      "6672\t7 * 953\tTrue\n",
      "6674\t6673\tFalse\n",
      "6676\t3 * 5^2 * 89\tFalse\n",
      "6678\t11 * 607\tFalse\n",
      "6680\t6679\tTrue\n",
      "6682\t3 * 17 * 131\tTrue\n",
      "6684\t41 * 163\tFalse\n",
      "6686\t5 * 7 * 191\tFalse\n",
      "6688\t3^2 * 743\tFalse\n",
      "6690\t6689\tTrue\n",
      "6692\t6691\tTrue\n",
      "6694\t3 * 23 * 97\tFalse\n",
      "6696\t5 * 13 * 103\tFalse\n",
      "6698\t37 * 181\tFalse\n",
      "6700\t3 * 7 * 11 * 29\tTrue\n",
      "6702\t6701\tTrue\n",
      "6704\t6703\tFalse\n",
      "6706\t3^2 * 5 * 149\tFalse\n",
      "6708\t19 * 353\tFalse\n",
      "6710\t6709\tTrue\n",
      "6712\t3 * 2237\tFalse\n",
      "6714\t7^2 * 137\tFalse\n",
      "6716\t5 * 17 * 79\tFalse\n",
      "6718\t3 * 2239\tFalse\n",
      "6720\t6719\tTrue\n",
      "6722\t11 * 13 * 47\tTrue\n",
      "6724\t3^4 * 83\tFalse\n",
      "6726\t5^2 * 269\tFalse\n",
      "6728\t7 * 31^2\tFalse\n",
      "6730\t3 * 2243\tFalse\n",
      "6732\t53 * 127\tTrue\n",
      "6734\t6733\tFalse\n",
      "6736\t3 * 5 * 449\tFalse\n",
      "6738\t6737\tFalse\n",
      "6740\t23 * 293\tTrue\n",
      "6742\t3^2 * 7 * 107\tFalse\n",
      "6744\t11 * 613\tFalse\n",
      "6746\t5 * 19 * 71\tFalse\n",
      "6748\t3 * 13 * 173\tFalse\n",
      "6750\t17 * 397\tTrue\n",
      "6752\t43 * 157\tTrue\n",
      "6754\t3 * 2251\tFalse\n",
      "6756\t5 * 7 * 193\tFalse\n",
      "6758\t29 * 233\tFalse\n",
      "6760\t3^2 * 751\tFalse\n",
      "6762\t6761\tTrue\n",
      "6764\t6763\tFalse\n",
      "6766\t3 * 5 * 11 * 41\tFalse\n",
      "6768\t67 * 101\tFalse\n",
      "6770\t7 * 967\tTrue\n",
      "6772\t3 * 37 * 61\tFalse\n",
      "6774\t13 * 521\tFalse\n",
      "6776\t5^2 * 271\tFalse\n",
      "6778\t3^3 * 251\tFalse\n",
      "6780\t6779\tTrue\n",
      "6782\t6781\tTrue\n",
      "6784\t3 * 7 * 17 * 19\tFalse\n",
      "6786\t5 * 23 * 59\tFalse\n",
      "6788\t11 * 617\tFalse\n",
      "6790\t3 * 31 * 73\tTrue\n",
      "6792\t6791\tTrue\n",
      "6794\t6793\tFalse\n",
      "6796\t3^2 * 5 * 151\tFalse\n",
      "6798\t7 * 971\tFalse\n",
      "6800\t13 * 523\tTrue\n",
      "6802\t3 * 2267\tFalse\n",
      "6804\t6803\tFalse\n",
      "6806\t5 * 1361\tFalse\n",
      "6808\t3 * 2269\tFalse\n",
      "6810\t11 * 619\tTrue\n",
      "6812\t7^2 * 139\tTrue\n",
      "6814\t3^2 * 757\tFalse\n",
      "6816\t5 * 29 * 47\tFalse\n",
      "6818\t17 * 401\tFalse\n",
      "6820\t3 * 2273\tFalse\n",
      "6822\t19 * 359\tTrue\n",
      "6824\t6823\tFalse\n",
      "6826\t3 * 5^2 * 7 * 13\tFalse\n",
      "6828\t6827\tFalse\n",
      "6830\t6829\tTrue\n",
      "6832\t3^3 * 11 * 23\tTrue\n",
      "6834\t6833\tFalse\n",
      "6836\t5 * 1367\tFalse\n",
      "6838\t3 * 43 * 53\tFalse\n",
      "6840\t7 * 977\tTrue\n",
      "6842\t6841\tTrue\n",
      "6844\t3 * 2281\tFalse\n",
      "6846\t5 * 37^2\tFalse\n",
      "6848\t41 * 167\tFalse\n",
      "6850\t3^2 * 761\tFalse\n",
      "6852\t13 * 17 * 31\tTrue\n",
      "6854\t7 * 11 * 89\tFalse\n",
      "6856\t3 * 5 * 457\tFalse\n",
      "6858\t6857\tFalse\n",
      "6860\t19^3\tTrue\n",
      "6862\t3 * 2287\tFalse\n",
      "6864\t6863\tFalse\n",
      "6866\t5 * 1373\tFalse\n",
      "6868\t3^2 * 7 * 109\tFalse\n",
      "6870\t6869\tTrue\n",
      "6872\t6871\tTrue\n",
      "6874\t3 * 29 * 79\tFalse\n",
      "6876\t5^4 * 11\tFalse\n",
      "6878\t13 * 23^2\tFalse\n",
      "6880\t3 * 2293\tFalse\n",
      "6882\t7 * 983\tTrue\n",
      "6884\t6883\tFalse\n",
      "6886\t3^4 * 5 * 17\tFalse\n",
      "6888\t71 * 97\tFalse\n",
      "6890\t83^2\tTrue\n",
      "6892\t3 * 2297\tFalse\n",
      "6894\t61 * 113\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6896\t5 * 7 * 197\tFalse\n",
      "6898\t3 * 11^2 * 19\tFalse\n",
      "6900\t6899\tTrue\n",
      "6902\t67 * 103\tTrue\n",
      "6904\t3^2 * 13 * 59\tFalse\n",
      "6906\t5 * 1381\tFalse\n",
      "6908\t6907\tFalse\n",
      "6910\t3 * 7^2 * 47\tFalse\n",
      "6912\t6911\tTrue\n",
      "6914\t31 * 223\tFalse\n",
      "6916\t3 * 5 * 461\tFalse\n",
      "6918\t6917\tFalse\n",
      "6920\t11 * 17 * 37\tTrue\n",
      "6922\t3^2 * 769\tFalse\n",
      "6924\t7 * 23 * 43\tFalse\n",
      "6926\t5^2 * 277\tFalse\n",
      "6928\t3 * 2309\tFalse\n",
      "6930\t13^2 * 41\tTrue\n",
      "6932\t29 * 239\tTrue\n",
      "6934\t3 * 2311\tFalse\n",
      "6936\t5 * 19 * 73\tFalse\n",
      "6938\t7 * 991\tFalse\n",
      "6940\t3^3 * 257\tFalse\n",
      "6942\t11 * 631\tTrue\n",
      "6944\t53 * 131\tFalse\n",
      "6946\t3 * 5 * 463\tFalse\n",
      "6948\t6947\tFalse\n",
      "6950\t6949\tTrue\n",
      "6952\t3 * 7 * 331\tFalse\n",
      "6954\t17 * 409\tFalse\n",
      "6956\t5 * 13 * 107\tFalse\n",
      "6958\t3^2 * 773\tFalse\n",
      "6960\t6959\tTrue\n",
      "6962\t6961\tTrue\n",
      "6964\t3 * 11 * 211\tFalse\n",
      "6966\t5 * 7 * 199\tFalse\n",
      "6968\t6967\tFalse\n",
      "6970\t3 * 23 * 101\tFalse\n",
      "6972\t6971\tTrue\n",
      "6974\t19 * 367\tFalse\n",
      "6976\t3^2 * 5^2 * 31\tFalse\n",
      "6978\t6977\tFalse\n",
      "6980\t7 * 997\tTrue\n",
      "6982\t3 * 13 * 179\tTrue\n",
      "6984\t6983\tFalse\n",
      "6986\t5 * 11 * 127\tFalse\n",
      "6988\t3 * 17 * 137\tFalse\n",
      "6990\t29 * 241\tTrue\n",
      "6992\t6991\tTrue\n",
      "6994\t3^3 * 7 * 37\tFalse\n",
      "6996\t5 * 1399\tFalse\n",
      "6998\t6997\tFalse\n",
      "7000\t3 * 2333\tFalse\n",
      "7002\t7001\tTrue\n",
      "7004\t47 * 149\tFalse\n",
      "7006\t3 * 5 * 467\tFalse\n",
      "7008\t7^2 * 11 * 13\tFalse\n",
      "7010\t43 * 163\tTrue\n",
      "7012\t3^2 * 19 * 41\tFalse\n",
      "7014\t7013\tFalse\n",
      "7016\t5 * 23 * 61\tFalse\n",
      "7018\t3 * 2339\tFalse\n",
      "7020\t7019\tTrue\n",
      "7022\t7 * 17 * 59\tTrue\n",
      "7024\t3 * 2341\tFalse\n",
      "7026\t5^2 * 281\tFalse\n",
      "7028\t7027\tFalse\n",
      "7030\t3^2 * 11 * 71\tTrue\n",
      "7032\t79 * 89\tTrue\n",
      "7034\t13 * 541\tFalse\n",
      "7036\t3 * 5 * 7 * 67\tFalse\n",
      "7038\t31 * 227\tFalse\n",
      "7040\t7039\tTrue\n",
      "7042\t3 * 2347\tFalse\n",
      "7044\t7043\tFalse\n",
      "7046\t5 * 1409\tFalse\n",
      "7048\t3^5 * 29\tFalse\n",
      "7050\t7 * 19 * 53\tTrue\n",
      "7052\t11 * 641\tTrue\n",
      "7054\t3 * 2351\tFalse\n",
      "7056\t5 * 17 * 83\tFalse\n",
      "7058\t7057\tFalse\n",
      "7060\t3 * 13 * 181\tFalse\n",
      "7062\t23 * 307\tTrue\n",
      "7064\t7 * 1009\tFalse\n",
      "7066\t3^2 * 5 * 157\tFalse\n",
      "7068\t37 * 191\tFalse\n",
      "7070\t7069\tTrue\n",
      "7072\t3 * 2357\tFalse\n",
      "7074\t11 * 643\tFalse\n",
      "7076\t5^2 * 283\tFalse\n",
      "7078\t3 * 7 * 337\tFalse\n",
      "7080\t7079\tTrue\n",
      "7082\t73 * 97\tTrue\n",
      "7084\t3^2 * 787\tFalse\n",
      "7086\t5 * 13 * 109\tFalse\n",
      "7088\t19 * 373\tFalse\n",
      "7090\t3 * 17 * 139\tTrue\n",
      "7092\t7 * 1013\tTrue\n",
      "7094\t41 * 173\tFalse\n",
      "7096\t3 * 5 * 11 * 43\tFalse\n",
      "7098\t47 * 151\tFalse\n",
      "7100\t31 * 229\tTrue\n",
      "7102\t3^3 * 263\tFalse\n",
      "7104\t7103\tFalse\n",
      "7106\t5 * 7^2 * 29\tFalse\n",
      "7108\t3 * 23 * 103\tFalse\n",
      "7110\t7109\tTrue\n",
      "7112\t13 * 547\tTrue\n",
      "7114\t3 * 2371\tFalse\n",
      "7116\t5 * 1423\tFalse\n",
      "7118\t11 * 647\tFalse\n",
      "7120\t3^2 * 7 * 113\tFalse\n",
      "7122\t7121\tTrue\n",
      "7124\t17 * 419\tFalse\n",
      "7126\t3 * 5^3 * 19\tFalse\n",
      "7128\t7127\tFalse\n",
      "7130\t7129\tTrue\n",
      "7132\t3 * 2377\tFalse\n",
      "7134\t7 * 1019\tFalse\n",
      "7136\t5 * 1427\tFalse\n",
      "7138\t3^2 * 13 * 61\tFalse\n",
      "7140\t11^2 * 59\tTrue\n",
      "7142\t37 * 193\tTrue\n",
      "7144\t3 * 2381\tFalse\n",
      "7146\t5 * 1429\tFalse\n",
      "7148\t7 * 1021\tFalse\n",
      "7150\t3 * 2383\tFalse\n",
      "7152\t7151\tTrue\n",
      "7154\t23 * 311\tFalse\n",
      "7156\t3^3 * 5 * 53\tFalse\n",
      "7158\t17 * 421\tFalse\n",
      "7160\t7159\tTrue\n",
      "7162\t3 * 7 * 11 * 31\tTrue\n",
      "7164\t13 * 19 * 29\tFalse\n",
      "7166\t5 * 1433\tFalse\n",
      "7168\t3 * 2389\tFalse\n",
      "7170\t67 * 107\tTrue\n",
      "7172\t71 * 101\tTrue\n",
      "7174\t3^2 * 797\tFalse\n",
      "7176\t5^2 * 7 * 41\tFalse\n",
      "7178\t7177\tFalse\n",
      "7180\t3 * 2393\tFalse\n",
      "7182\t43 * 167\tTrue\n",
      "7184\t11 * 653\tFalse\n",
      "7186\t3 * 5 * 479\tFalse\n",
      "7188\t7187\tFalse\n",
      "7190\t7 * 13 * 79\tTrue\n",
      "7192\t3^2 * 17 * 47\tFalse\n",
      "7194\t7193\tFalse\n",
      "7196\t5 * 1439\tFalse\n",
      "7198\t3 * 2399\tFalse\n",
      "7200\t23 * 313\tTrue\n",
      "7202\t19 * 379\tTrue\n",
      "7204\t3 * 7^4\tFalse\n",
      "7206\t5 * 11 * 131\tFalse\n",
      "7208\t7207\tFalse\n",
      "7210\t3^4 * 89\tFalse\n",
      "7212\t7211\tTrue\n",
      "7214\t7213\tFalse\n",
      "7216\t3 * 5 * 13 * 37\tFalse\n",
      "7218\t7 * 1031\tFalse\n",
      "7220\t7219\tTrue\n",
      "7222\t3 * 29 * 83\tFalse\n",
      "7224\t31 * 233\tFalse\n",
      "7226\t5^2 * 17^2\tFalse\n",
      "7228\t3^2 * 11 * 73\tFalse\n",
      "7230\t7229\tTrue\n",
      "7232\t7 * 1033\tTrue\n",
      "7234\t3 * 2411\tFalse\n",
      "7236\t5 * 1447\tFalse\n",
      "7238\t7237\tFalse\n",
      "7240\t3 * 19 * 127\tFalse\n",
      "7242\t13 * 557\tTrue\n",
      "7244\t7243\tFalse\n",
      "7246\t3^2 * 5 * 7 * 23\tFalse\n",
      "7248\t7247\tFalse\n",
      "7250\t11 * 659\tTrue\n",
      "7252\t3 * 2417\tFalse\n",
      "7254\t7253\tFalse\n",
      "7256\t5 * 1451\tFalse\n",
      "7258\t3 * 41 * 59\tFalse\n",
      "7260\t7 * 17 * 61\tTrue\n",
      "7262\t53 * 137\tTrue\n",
      "7264\t3^3 * 269\tFalse\n",
      "7266\t5 * 1453\tFalse\n",
      "7268\t13^2 * 43\tFalse\n",
      "7270\t3 * 2423\tFalse\n",
      "7272\t11 * 661\tTrue\n",
      "7274\t7 * 1039\tFalse\n",
      "7276\t3 * 5^2 * 97\tFalse\n",
      "7278\t19 * 383\tFalse\n",
      "7280\t29 * 251\tTrue\n",
      "7282\t3^2 * 809\tFalse\n",
      "7284\t7283\tFalse\n",
      "7286\t5 * 31 * 47\tFalse\n",
      "7288\t3 * 7 * 347\tFalse\n",
      "7290\t37 * 197\tTrue\n",
      "7292\t23 * 317\tTrue\n",
      "7294\t3 * 11 * 13 * 17\tFalse\n",
      "7296\t5 * 1459\tFalse\n",
      "7298\t7297\tFalse\n",
      "7300\t3^2 * 811\tFalse\n",
      "7302\t7^2 * 149\tTrue\n",
      "7304\t67 * 109\tFalse\n",
      "7306\t3 * 5 * 487\tFalse\n",
      "7308\t7307\tFalse\n",
      "7310\t7309\tTrue\n",
      "7312\t3 * 2437\tFalse\n",
      "7314\t71 * 103\tFalse\n",
      "7316\t5 * 7 * 11 * 19\tFalse\n",
      "7318\t3^3 * 271\tFalse\n",
      "7320\t13 * 563\tTrue\n",
      "7322\t7321\tTrue\n",
      "7324\t3 * 2441\tFalse\n",
      "7326\t5^2 * 293\tFalse\n",
      "7328\t17 * 431\tFalse\n",
      "7330\t3 * 7 * 349\tFalse\n",
      "7332\t7331\tTrue\n",
      "7334\t7333\tFalse\n",
      "7336\t3^2 * 5 * 163\tFalse\n",
      "7338\t11 * 23 * 29\tFalse\n",
      "7340\t41 * 179\tTrue\n",
      "7342\t3 * 2447\tFalse\n",
      "7344\t7 * 1049\tFalse\n",
      "7346\t5 * 13 * 113\tFalse\n",
      "7348\t3 * 31 * 79\tFalse\n",
      "7350\t7349\tTrue\n",
      "7352\t7351\tTrue\n",
      "7354\t3^2 * 19 * 43\tFalse\n",
      "7356\t5 * 1471\tFalse\n",
      "7358\t7 * 1051\tFalse\n",
      "7360\t3 * 11 * 223\tTrue\n",
      "7362\t17 * 433\tTrue\n",
      "7364\t37 * 199\tFalse\n",
      "7366\t3 * 5 * 491\tFalse\n",
      "7368\t53 * 139\tFalse\n",
      "7370\t7369\tTrue\n",
      "7372\t3^4 * 7 * 13\tFalse\n",
      "7374\t73 * 101\tFalse\n",
      "7376\t5^3 * 59\tFalse\n",
      "7378\t3 * 2459\tFalse\n",
      "7380\t47 * 157\tTrue\n",
      "7382\t11^2 * 61\tTrue\n",
      "7384\t3 * 23 * 107\tFalse\n",
      "7386\t5 * 7 * 211\tFalse\n",
      "7388\t83 * 89\tFalse\n",
      "7390\t3^2 * 821\tFalse\n",
      "7392\t19 * 389\tTrue\n",
      "7394\t7393\tFalse\n",
      "7396\t3 * 5 * 17 * 29\tFalse\n",
      "7398\t13 * 569\tFalse\n",
      "7400\t7^2 * 151\tTrue\n",
      "7402\t3 * 2467\tFalse\n",
      "7404\t11 * 673\tFalse\n",
      "7406\t5 * 1481\tFalse\n",
      "7408\t3^2 * 823\tFalse\n",
      "7410\t31 * 239\tTrue\n",
      "7412\t7411\tTrue\n",
      "7414\t3 * 7 * 353\tFalse\n",
      "7416\t5 * 1483\tFalse\n",
      "7418\t7417\tFalse\n",
      "7420\t3 * 2473\tFalse\n",
      "7422\t41 * 181\tTrue\n",
      "7424\t13 * 571\tFalse\n",
      "7426\t3^3 * 5^2 * 11\tFalse\n",
      "7428\t7 * 1061\tFalse\n",
      "7430\t17 * 19 * 23\tTrue\n",
      "7432\t3 * 2477\tFalse\n",
      "7434\t7433\tFalse\n",
      "7436\t5 * 1487\tFalse\n",
      "7438\t3 * 37 * 67\tFalse\n",
      "7440\t43 * 173\tTrue\n",
      "7442\t7 * 1063\tTrue\n",
      "7444\t3^2 * 827\tFalse\n",
      "7446\t5 * 1489\tFalse\n",
      "7448\t11 * 677\tFalse\n",
      "7450\t3 * 13 * 191\tTrue\n",
      "7452\t7451\tTrue\n",
      "7454\t29 * 257\tFalse\n",
      "7456\t3 * 5 * 7 * 71\tFalse\n",
      "7458\t7457\tFalse\n",
      "7460\t7459\tTrue\n",
      "7462\t3^2 * 829\tFalse\n",
      "7464\t17 * 439\tFalse\n",
      "7466\t5 * 1493\tFalse\n",
      "7468\t3 * 19 * 131\tFalse\n",
      "7470\t7 * 11 * 97\tTrue\n",
      "7472\t31 * 241\tTrue\n",
      "7474\t3 * 47 * 53\tFalse\n",
      "7476\t5^2 * 13 * 23\tFalse\n",
      "7478\t7477\tFalse\n",
      "7480\t3^3 * 277\tFalse\n",
      "7482\t7481\tTrue\n",
      "7484\t7 * 1069\tFalse\n",
      "7486\t3 * 5 * 499\tFalse\n",
      "7488\t7487\tFalse\n",
      "7490\t7489\tTrue\n",
      "7492\t3 * 11 * 227\tTrue\n",
      "7494\t59 * 127\tFalse\n",
      "7496\t5 * 1499\tFalse\n",
      "7498\t3^2 * 7^2 * 17\tFalse\n",
      "7500\t7499\tTrue\n",
      "7502\t13 * 577\tTrue\n",
      "7504\t3 * 41 * 61\tFalse\n",
      "7506\t5 * 19 * 79\tFalse\n",
      "7508\t7507\tFalse\n",
      "7510\t3 * 2503\tFalse\n",
      "7512\t7 * 29 * 37\tTrue\n",
      "7514\t11 * 683\tFalse\n",
      "7516\t3^2 * 5 * 167\tFalse\n",
      "7518\t7517\tFalse\n",
      "7520\t73 * 103\tTrue\n",
      "7522\t3 * 23 * 109\tFalse\n",
      "7524\t7523\tFalse\n",
      "7526\t5^2 * 7 * 43\tFalse\n",
      "7528\t3 * 13 * 193\tFalse\n",
      "7530\t7529\tTrue\n",
      "7532\t17 * 443\tTrue\n",
      "7534\t3^5 * 31\tFalse\n",
      "7536\t5 * 11 * 137\tFalse\n",
      "7538\t7537\tFalse\n",
      "7540\t3 * 7 * 359\tFalse\n",
      "7542\t7541\tTrue\n",
      "7544\t19 * 397\tFalse\n",
      "7546\t3 * 5 * 503\tFalse\n",
      "7548\t7547\tFalse\n",
      "7550\t7549\tTrue\n",
      "7552\t3^2 * 839\tFalse\n",
      "7554\t7 * 13 * 83\tFalse\n",
      "7556\t5 * 1511\tFalse\n",
      "7558\t3 * 11 * 229\tFalse\n",
      "7560\t7559\tTrue\n",
      "7562\t7561\tTrue\n",
      "7564\t3 * 2521\tFalse\n",
      "7566\t5 * 17 * 89\tFalse\n",
      "7568\t7 * 23 * 47\tFalse\n",
      "7570\t3^2 * 29^2\tFalse\n",
      "7572\t67 * 113\tTrue\n",
      "7574\t7573\tFalse\n",
      "7576\t3 * 5^2 * 101\tFalse\n",
      "7578\t7577\tFalse\n",
      "7580\t11 * 13 * 53\tTrue\n",
      "7582\t3 * 7 * 19^2\tFalse\n",
      "7584\t7583\tFalse\n",
      "7586\t5 * 37 * 41\tFalse\n",
      "7588\t3^3 * 281\tFalse\n",
      "7590\t7589\tTrue\n",
      "7592\t7591\tTrue\n",
      "7594\t3 * 2531\tFalse\n",
      "7596\t5 * 7^2 * 31\tFalse\n",
      "7598\t71 * 107\tFalse\n",
      "7600\t3 * 17 * 149\tFalse\n",
      "7602\t11 * 691\tTrue\n",
      "7604\t7603\tFalse\n",
      "7606\t3^2 * 5 * 13^2\tFalse\n",
      "7608\t7607\tFalse\n",
      "7610\t7 * 1087\tTrue\n",
      "7612\t3 * 43 * 59\tFalse\n",
      "7614\t23 * 331\tFalse\n",
      "7616\t5 * 1523\tFalse\n",
      "7618\t3 * 2539\tFalse\n",
      "7620\t19 * 401\tTrue\n",
      "7622\t7621\tTrue\n",
      "7624\t3^2 * 7 * 11^2\tFalse\n",
      "7626\t5^3 * 61\tFalse\n",
      "7628\t29 * 263\tFalse\n",
      "7630\t3 * 2543\tFalse\n",
      "7632\t13 * 587\tTrue\n",
      "7634\t17 * 449\tFalse\n",
      "7636\t3 * 5 * 509\tFalse\n",
      "7638\t7 * 1091\tFalse\n",
      "7640\t7639\tTrue\n",
      "7642\t3^3 * 283\tFalse\n",
      "7644\t7643\tFalse\n",
      "7646\t5 * 11 * 139\tFalse\n",
      "7648\t3 * 2549\tFalse\n",
      "7650\t7649\tTrue\n",
      "7652\t7 * 1093\tTrue\n",
      "7654\t3 * 2551\tFalse\n",
      "7656\t5 * 1531\tFalse\n",
      "7658\t13 * 19 * 31\tFalse\n",
      "7660\t3^2 * 23 * 37\tFalse\n",
      "7662\t47 * 163\tTrue\n",
      "7664\t79 * 97\tFalse\n",
      "7666\t3 * 5 * 7 * 73\tFalse\n",
      "7668\t11 * 17 * 41\tFalse\n",
      "7670\t7669\tTrue\n",
      "7672\t3 * 2557\tFalse\n",
      "7674\t7673\tFalse\n",
      "7676\t5^2 * 307\tFalse\n",
      "7678\t3^2 * 853\tFalse\n",
      "7680\t7 * 1097\tTrue\n",
      "7682\t7681\tTrue\n",
      "7684\t3 * 13 * 197\tFalse\n",
      "7686\t5 * 29 * 53\tFalse\n",
      "7688\t7687\tFalse\n",
      "7690\t3 * 11 * 233\tTrue\n",
      "7692\t7691\tTrue\n",
      "7694\t7^2 * 157\tFalse\n",
      "7696\t3^4 * 5 * 19\tFalse\n",
      "7698\t43 * 179\tFalse\n",
      "7700\t7699\tTrue\n",
      "7702\t3 * 17 * 151\tTrue\n",
      "7704\t7703\tFalse\n",
      "7706\t5 * 23 * 67\tFalse\n",
      "7708\t3 * 7 * 367\tFalse\n",
      "7710\t13 * 593\tTrue\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7712\t11 * 701\tTrue\n",
      "7714\t3^2 * 857\tFalse\n",
      "7716\t5 * 1543\tFalse\n",
      "7718\t7717\tFalse\n",
      "7720\t3 * 31 * 83\tFalse\n",
      "7722\t7 * 1103\tTrue\n",
      "7724\t7723\tFalse\n",
      "7726\t3 * 5^2 * 103\tFalse\n",
      "7728\t7727\tFalse\n",
      "7730\t59 * 131\tTrue\n",
      "7732\t3^2 * 859\tFalse\n",
      "7734\t11 * 19 * 37\tFalse\n",
      "7736\t5 * 7 * 13 * 17\tFalse\n",
      "7738\t3 * 2579\tFalse\n",
      "7740\t71 * 109\tTrue\n",
      "7742\t7741\tTrue\n",
      "7744\t3 * 29 * 89\tFalse\n",
      "7746\t5 * 1549\tFalse\n",
      "7748\t61 * 127\tFalse\n",
      "7750\t3^3 * 7 * 41\tFalse\n",
      "7752\t23 * 337\tTrue\n",
      "7754\t7753\tFalse\n",
      "7756\t3 * 5 * 11 * 47\tFalse\n",
      "7758\t7757\tFalse\n",
      "7760\t7759\tTrue\n",
      "7762\t3 * 13 * 199\tTrue\n",
      "7764\t7 * 1109\tFalse\n",
      "7766\t5 * 1553\tFalse\n",
      "7768\t3^2 * 863\tFalse\n",
      "7770\t17 * 457\tTrue\n",
      "7772\t19 * 409\tTrue\n",
      "7774\t3 * 2591\tFalse\n",
      "7776\t5^2 * 311\tFalse\n",
      "7778\t7 * 11 * 101\tFalse\n",
      "7780\t3 * 2593\tFalse\n",
      "7782\t31 * 251\tTrue\n",
      "7784\t43 * 181\tFalse\n",
      "7786\t3^2 * 5 * 173\tFalse\n",
      "7788\t13 * 599\tFalse\n",
      "7790\t7789\tTrue\n",
      "7792\t3 * 7^2 * 53\tFalse\n",
      "7794\t7793\tFalse\n",
      "7796\t5 * 1559\tFalse\n",
      "7798\t3 * 23 * 113\tFalse\n",
      "7800\t11 * 709\tTrue\n",
      "7802\t29 * 269\tTrue\n",
      "7804\t3^3 * 17^2\tFalse\n",
      "7806\t5 * 7 * 223\tFalse\n",
      "7808\t37 * 211\tFalse\n",
      "7810\t3 * 19 * 137\tTrue\n",
      "7812\t73 * 107\tTrue\n",
      "7814\t13 * 601\tFalse\n",
      "7816\t3 * 5 * 521\tFalse\n",
      "7818\t7817\tFalse\n",
      "7820\t7 * 1117\tTrue\n",
      "7822\t3^2 * 11 * 79\tTrue\n",
      "7824\t7823\tFalse\n",
      "7826\t5^2 * 313\tFalse\n",
      "7828\t3 * 2609\tFalse\n",
      "7830\t7829\tTrue\n",
      "7832\t41 * 191\tTrue\n",
      "7834\t3 * 7 * 373\tFalse\n",
      "7836\t5 * 1567\tFalse\n",
      "7838\t17 * 461\tFalse\n",
      "7840\t3^2 * 13 * 67\tTrue\n",
      "7842\t7841\tTrue\n",
      "7844\t11 * 23 * 31\tFalse\n",
      "7846\t3 * 5 * 523\tFalse\n",
      "7848\t7 * 19 * 59\tFalse\n",
      "7850\t47 * 167\tTrue\n",
      "7852\t3 * 2617\tFalse\n",
      "7854\t7853\tFalse\n",
      "7856\t5 * 1571\tFalse\n",
      "7858\t3^4 * 97\tFalse\n",
      "7860\t29 * 271\tTrue\n",
      "7862\t7 * 1123\tTrue\n",
      "7864\t3 * 2621\tFalse\n",
      "7866\t5 * 11^2 * 13\tFalse\n",
      "7868\t7867\tFalse\n",
      "7870\t3 * 43 * 61\tFalse\n",
      "7872\t17 * 463\tTrue\n",
      "7874\t7873\tFalse\n",
      "7876\t3^2 * 5^3 * 7\tFalse\n",
      "7878\t7877\tFalse\n",
      "7880\t7879\tTrue\n",
      "7882\t3 * 37 * 71\tTrue\n",
      "7884\t7883\tFalse\n",
      "7886\t5 * 19 * 83\tFalse\n",
      "7888\t3 * 11 * 239\tFalse\n",
      "7890\t7^3 * 23\tTrue\n",
      "7892\t13 * 607\tTrue\n",
      "7894\t3^2 * 877\tFalse\n",
      "7896\t5 * 1579\tFalse\n",
      "7898\t53 * 149\tFalse\n",
      "7900\t3 * 2633\tFalse\n",
      "7902\t7901\tTrue\n",
      "7904\t7 * 1129\tFalse\n",
      "7906\t3 * 5 * 17 * 31\tFalse\n",
      "7908\t7907\tFalse\n",
      "7910\t11 * 719\tTrue\n",
      "7912\t3^3 * 293\tFalse\n",
      "7914\t41 * 193\tFalse\n",
      "7916\t5 * 1583\tFalse\n",
      "7918\t3 * 7 * 13 * 29\tFalse\n",
      "7920\t7919\tTrue\n",
      "7922\t89^2\tTrue\n",
      "7924\t3 * 19 * 139\tFalse\n",
      "7926\t5^2 * 317\tFalse\n",
      "7928\t7927\tFalse\n",
      "7930\t3^2 * 881\tFalse\n",
      "7932\t7 * 11 * 103\tTrue\n",
      "7934\t7933\tFalse\n",
      "7936\t3 * 5 * 23^2\tFalse\n",
      "7938\t7937\tFalse\n",
      "7940\t17 * 467\tTrue\n",
      "7942\t3 * 2647\tFalse\n",
      "7944\t13^2 * 47\tFalse\n",
      "7946\t5 * 7 * 227\tFalse\n",
      "7948\t3^2 * 883\tFalse\n",
      "7950\t7949\tTrue\n",
      "7952\t7951\tTrue\n",
      "7954\t3 * 11 * 241\tFalse\n",
      "7956\t5 * 37 * 43\tFalse\n",
      "7958\t73 * 109\tFalse\n",
      "7960\t3 * 7 * 379\tFalse\n",
      "7962\t19 * 419\tTrue\n",
      "7964\t7963\tFalse\n",
      "7966\t3^3 * 5 * 59\tFalse\n",
      "7968\t31 * 257\tFalse\n",
      "7970\t13 * 613\tTrue\n",
      "7972\t3 * 2657\tFalse\n",
      "7974\t7 * 17 * 67\tFalse\n",
      "7976\t5^2 * 11 * 29\tFalse\n",
      "7978\t3 * 2659\tFalse\n",
      "7980\t79 * 101\tTrue\n",
      "7982\t23 * 347\tTrue\n",
      "7984\t3^2 * 887\tFalse\n",
      "7986\t5 * 1597\tFalse\n",
      "7988\t7^2 * 163\tFalse\n",
      "7990\t3 * 2663\tFalse\n",
      "7992\t61 * 131\tTrue\n",
      "7994\t7993\tFalse\n",
      "7996\t3 * 5 * 13 * 41\tFalse\n",
      "7998\t11 * 727\tFalse\n",
      "8000\t19 * 421\tTrue\n",
      "8002\t3^2 * 7 * 127\tFalse\n",
      "8004\t53 * 151\tFalse\n",
      "8006\t5 * 1601\tFalse\n",
      "8008\t3 * 17 * 157\tFalse\n",
      "8010\t8009\tTrue\n",
      "8012\t8011\tTrue\n",
      "8014\t3 * 2671\tFalse\n",
      "8016\t5 * 7 * 229\tFalse\n",
      "8018\t8017\tFalse\n",
      "8020\t3^6 * 11\tTrue\n",
      "8022\t13 * 617\tTrue\n",
      "8024\t71 * 113\tFalse\n",
      "8026\t3 * 5^2 * 107\tFalse\n",
      "8028\t23 * 349\tFalse\n",
      "8030\t7 * 31 * 37\tTrue\n",
      "8032\t3 * 2677\tFalse\n",
      "8034\t29 * 277\tFalse\n",
      "8036\t5 * 1607\tFalse\n",
      "8038\t3^2 * 19 * 47\tFalse\n",
      "8040\t8039\tTrue\n",
      "8042\t11 * 17 * 43\tTrue\n",
      "8044\t3 * 7 * 383\tFalse\n",
      "8046\t5 * 1609\tFalse\n",
      "8048\t13 * 619\tFalse\n",
      "8050\t3 * 2683\tFalse\n",
      "8052\t83 * 97\tTrue\n",
      "8054\t8053\tFalse\n",
      "8056\t3^2 * 5 * 179\tFalse\n",
      "8058\t7 * 1151\tFalse\n",
      "8060\t8059\tTrue\n",
      "8062\t3 * 2687\tFalse\n",
      "8064\t11 * 733\tFalse\n",
      "8066\t5 * 1613\tFalse\n",
      "8068\t3 * 2689\tFalse\n",
      "8070\t8069\tTrue\n",
      "8072\t7 * 1153\tTrue\n",
      "8074\t3^3 * 13 * 23\tFalse\n",
      "8076\t5^2 * 17 * 19\tFalse\n",
      "8078\t41 * 197\tFalse\n",
      "8080\t3 * 2693\tFalse\n",
      "8082\t8081\tTrue\n",
      "8084\t59 * 137\tFalse\n",
      "8086\t3 * 5 * 7^2 * 11\tFalse\n",
      "8088\t8087\tFalse\n",
      "8090\t8089\tTrue\n",
      "8092\t3^2 * 29 * 31\tFalse\n",
      "8094\t8093\tFalse\n",
      "8096\t5 * 1619\tFalse\n",
      "8098\t3 * 2699\tFalse\n",
      "8100\t7 * 13 * 89\tTrue\n",
      "8102\t8101\tTrue\n",
      "8104\t3 * 37 * 73\tFalse\n",
      "8106\t5 * 1621\tFalse\n",
      "8108\t11^2 * 67\tFalse\n",
      "8110\t3^2 * 17 * 53\tTrue\n",
      "8112\t8111\tTrue\n",
      "8114\t7 * 19 * 61\tFalse\n",
      "8116\t3 * 5 * 541\tFalse\n",
      "8118\t8117\tFalse\n",
      "8120\t23 * 353\tTrue\n",
      "8122\t3 * 2707\tFalse\n",
      "8124\t8123\tFalse\n",
      "8126\t5^4 * 13\tFalse\n",
      "8128\t3^3 * 7 * 43\tFalse\n",
      "8130\t11 * 739\tTrue\n",
      "8132\t47 * 173\tTrue\n",
      "8134\t3 * 2711\tFalse\n",
      "8136\t5 * 1627\tFalse\n",
      "8138\t79 * 103\tFalse\n",
      "8140\t3 * 2713\tFalse\n",
      "8142\t7 * 1163\tTrue\n",
      "8144\t17 * 479\tFalse\n",
      "8146\t3^2 * 5 * 181\tFalse\n",
      "8148\t8147\tFalse\n",
      "8150\t29 * 281\tTrue\n",
      "8152\t3 * 11 * 13 * 19\tFalse\n",
      "8154\t31 * 263\tFalse\n",
      "8156\t5 * 7 * 233\tFalse\n",
      "8158\t3 * 2719\tFalse\n",
      "8160\t41 * 199\tTrue\n",
      "8162\t8161\tTrue\n",
      "8164\t3^2 * 907\tFalse\n",
      "8166\t5 * 23 * 71\tFalse\n",
      "8168\t8167\tFalse\n",
      "8170\t3 * 7 * 389\tFalse\n",
      "8172\t8171\tTrue\n",
      "8174\t11 * 743\tFalse\n",
      "8176\t3 * 5^2 * 109\tFalse\n",
      "8178\t13 * 17 * 37\tFalse\n",
      "8180\t8179\tTrue\n",
      "8182\t3^4 * 101\tFalse\n",
      "8184\t7^2 * 167\tFalse\n",
      "8186\t5 * 1637\tFalse\n",
      "8188\t3 * 2729\tFalse\n",
      "8190\t19 * 431\tTrue\n",
      "8192\t8191\tTrue\n",
      "8194\t3 * 2731\tFalse\n",
      "8196\t5 * 11 * 149\tFalse\n",
      "8198\t7 * 1171\tFalse\n",
      "8200\t3^2 * 911\tFalse\n",
      "8202\t59 * 139\tTrue\n",
      "8204\t13 * 631\tFalse\n",
      "8206\t3 * 5 * 547\tFalse\n",
      "8208\t29 * 283\tFalse\n",
      "8210\t8209\tTrue\n",
      "8212\t3 * 7 * 17 * 23\tFalse\n",
      "8214\t43 * 191\tFalse\n",
      "8216\t5 * 31 * 53\tFalse\n",
      "8218\t3^2 * 11 * 83\tFalse\n",
      "8220\t8219\tTrue\n",
      "8222\t8221\tTrue\n",
      "8224\t3 * 2741\tFalse\n",
      "8226\t5^2 * 7 * 47\tFalse\n",
      "8228\t19 * 433\tFalse\n",
      "8230\t3 * 13 * 211\tTrue\n",
      "8232\t8231\tTrue\n",
      "8234\t8233\tFalse\n",
      "8236\t3^3 * 5 * 61\tFalse\n",
      "8238\t8237\tFalse\n",
      "8240\t7 * 11 * 107\tTrue\n",
      "8242\t3 * 41 * 67\tFalse\n",
      "8244\t8243\tFalse\n",
      "8246\t5 * 17 * 97\tFalse\n",
      "8248\t3 * 2749\tFalse\n",
      "8250\t73 * 113\tTrue\n",
      "8252\t37 * 223\tTrue\n",
      "8254\t3^2 * 7 * 131\tFalse\n",
      "8256\t5 * 13 * 127\tFalse\n",
      "8258\t23 * 359\tFalse\n",
      "8260\t3 * 2753\tFalse\n",
      "8262\t11 * 751\tTrue\n",
      "8264\t8263\tFalse\n",
      "8266\t3 * 5 * 19 * 29\tFalse\n",
      "8268\t7 * 1181\tFalse\n",
      "8270\t8269\tTrue\n",
      "8272\t3^2 * 919\tFalse\n",
      "8274\t8273\tFalse\n",
      "8276\t5^2 * 331\tFalse\n",
      "8278\t3 * 31 * 89\tFalse\n",
      "8280\t17 * 487\tTrue\n",
      "8282\t7^2 * 13^2\tTrue\n",
      "8284\t3 * 11 * 251\tFalse\n",
      "8286\t5 * 1657\tFalse\n",
      "8288\t8287\tFalse\n",
      "8290\t3^3 * 307\tFalse\n",
      "8292\t8291\tTrue\n",
      "8294\t8293\tFalse\n",
      "8296\t3 * 5 * 7 * 79\tFalse\n",
      "8298\t8297\tFalse\n",
      "8300\t43 * 193\tTrue\n",
      "8302\t3 * 2767\tFalse\n",
      "8304\t19^2 * 23\tFalse\n",
      "8306\t5 * 11 * 151\tFalse\n",
      "8308\t3^2 * 13 * 71\tFalse\n",
      "8310\t7 * 1187\tTrue\n",
      "8312\t8311\tTrue\n",
      "8314\t3 * 17 * 163\tFalse\n",
      "8316\t5 * 1663\tFalse\n",
      "8318\t8317\tFalse\n",
      "8320\t3 * 47 * 59\tFalse\n",
      "8322\t53 * 157\tTrue\n",
      "8324\t7 * 29 * 41\tFalse\n",
      "8326\t3^2 * 5^2 * 37\tFalse\n",
      "8328\t11 * 757\tFalse\n",
      "8330\t8329\tTrue\n",
      "8332\t3 * 2777\tFalse\n",
      "8334\t13 * 641\tFalse\n",
      "8336\t5 * 1667\tFalse\n",
      "8338\t3 * 7 * 397\tFalse\n",
      "8340\t31 * 269\tTrue\n",
      "8342\t19 * 439\tTrue\n",
      "8344\t3^4 * 103\tFalse\n",
      "8346\t5 * 1669\tFalse\n",
      "8348\t17 * 491\tFalse\n",
      "8350\t3 * 11^2 * 23\tFalse\n",
      "8352\t7 * 1193\tTrue\n",
      "8354\t8353\tFalse\n",
      "8356\t3 * 5 * 557\tFalse\n",
      "8358\t61 * 137\tFalse\n",
      "8360\t13 * 643\tTrue\n",
      "8362\t3^2 * 929\tFalse\n",
      "8364\t8363\tFalse\n",
      "8366\t5 * 7 * 239\tFalse\n",
      "8368\t3 * 2789\tFalse\n",
      "8370\t8369\tTrue\n",
      "8372\t11 * 761\tTrue\n",
      "8374\t3 * 2791\tFalse\n",
      "8376\t5^3 * 67\tFalse\n",
      "8378\t8377\tFalse\n",
      "8380\t3^2 * 7^2 * 19\tTrue\n",
      "8382\t17^2 * 29\tTrue\n",
      "8384\t83 * 101\tFalse\n",
      "8386\t3 * 5 * 13 * 43\tFalse\n",
      "8388\t8387\tFalse\n",
      "8390\t8389\tTrue\n",
      "8392\t3 * 2797\tFalse\n",
      "8394\t7 * 11 * 109\tFalse\n",
      "8396\t5 * 23 * 73\tFalse\n",
      "8398\t3^3 * 311\tFalse\n",
      "8400\t37 * 227\tTrue\n",
      "8402\t31 * 271\tTrue\n",
      "8404\t3 * 2801\tFalse\n",
      "8406\t5 * 41^2\tFalse\n",
      "8408\t7 * 1201\tFalse\n",
      "8410\t3 * 2803\tFalse\n",
      "8412\t13 * 647\tTrue\n",
      "8414\t47 * 179\tFalse\n",
      "8416\t3^2 * 5 * 11 * 17\tFalse\n",
      "8418\t19 * 443\tFalse\n",
      "8420\t8419\tTrue\n",
      "8422\t3 * 7 * 401\tFalse\n",
      "8424\t8423\tFalse\n",
      "8426\t5^2 * 337\tFalse\n",
      "8428\t3 * 53^2\tFalse\n",
      "8430\t8429\tTrue\n",
      "8432\t8431\tTrue\n",
      "8434\t3^2 * 937\tFalse\n",
      "8436\t5 * 7 * 241\tFalse\n",
      "8438\t11 * 13 * 59\tFalse\n",
      "8440\t3 * 29 * 97\tFalse\n",
      "8442\t23 * 367\tTrue\n",
      "8444\t8443\tFalse\n",
      "8446\t3 * 5 * 563\tFalse\n",
      "8448\t8447\tFalse\n",
      "8450\t7 * 17 * 71\tTrue\n",
      "8452\t3^3 * 313\tFalse\n",
      "8454\t79 * 107\tFalse\n",
      "8456\t5 * 19 * 89\tFalse\n",
      "8458\t3 * 2819\tFalse\n",
      "8460\t11 * 769\tTrue\n",
      "8462\t8461\tTrue\n",
      "8464\t3 * 7 * 13 * 31\tFalse\n",
      "8466\t5 * 1693\tFalse\n",
      "8468\t8467\tFalse\n",
      "8470\t3^2 * 941\tFalse\n",
      "8472\t43 * 197\tTrue\n",
      "8474\t37 * 229\tFalse\n",
      "8476\t3 * 5^2 * 113\tFalse\n",
      "8478\t7^2 * 173\tFalse\n",
      "8480\t61 * 139\tTrue\n",
      "8482\t3 * 11 * 257\tTrue\n",
      "8484\t17 * 499\tFalse\n",
      "8486\t5 * 1697\tFalse\n",
      "8488\t3^2 * 23 * 41\tFalse\n",
      "8490\t13 * 653\tTrue\n",
      "8492\t7 * 1213\tTrue\n",
      "8494\t3 * 19 * 149\tFalse\n",
      "8496\t5 * 1699\tFalse\n",
      "8498\t29 * 293\tFalse\n",
      "8500\t3 * 2833\tFalse\n",
      "8502\t8501\tTrue\n",
      "8504\t11 * 773\tFalse\n",
      "8506\t3^5 * 5 * 7\tFalse\n",
      "8508\t47 * 181\tFalse\n",
      "8510\t67 * 127\tTrue\n",
      "8512\t3 * 2837\tFalse\n",
      "8514\t8513\tFalse\n",
      "8516\t5 * 13 * 131\tFalse\n",
      "8518\t3 * 17 * 167\tFalse\n",
      "8520\t7 * 1217\tTrue\n",
      "8522\t8521\tTrue\n",
      "8524\t3^2 * 947\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8526\t5^2 * 11 * 31\tFalse\n",
      "8528\t8527\tFalse\n",
      "8530\t3 * 2843\tFalse\n",
      "8532\t19 * 449\tTrue\n",
      "8534\t7 * 23 * 53\tFalse\n",
      "8536\t3 * 5 * 569\tFalse\n",
      "8538\t8537\tFalse\n",
      "8540\t8539\tTrue\n",
      "8542\t3^2 * 13 * 73\tTrue\n",
      "8544\t8543\tFalse\n",
      "8546\t5 * 1709\tFalse\n",
      "8548\t3 * 7 * 11 * 37\tFalse\n",
      "8550\t83 * 103\tTrue\n",
      "8552\t17 * 503\tTrue\n",
      "8554\t3 * 2851\tFalse\n",
      "8556\t5 * 29 * 59\tFalse\n",
      "8558\t43 * 199\tFalse\n",
      "8560\t3^3 * 317\tFalse\n",
      "8562\t7 * 1223\tTrue\n",
      "8564\t8563\tFalse\n",
      "8566\t3 * 5 * 571\tFalse\n",
      "8568\t13 * 659\tFalse\n",
      "8570\t11 * 19 * 41\tTrue\n",
      "8572\t3 * 2857\tFalse\n",
      "8574\t8573\tFalse\n",
      "8576\t5^2 * 7^3\tFalse\n",
      "8578\t3^2 * 953\tFalse\n",
      "8580\t23 * 373\tTrue\n",
      "8582\t8581\tTrue\n",
      "8584\t3 * 2861\tFalse\n",
      "8586\t5 * 17 * 101\tFalse\n",
      "8588\t31 * 277\tFalse\n",
      "8590\t3 * 7 * 409\tFalse\n",
      "8592\t11^2 * 71\tTrue\n",
      "8594\t13 * 661\tFalse\n",
      "8596\t3^2 * 5 * 191\tFalse\n",
      "8598\t8597\tFalse\n",
      "8600\t8599\tTrue\n",
      "8602\t3 * 47 * 61\tFalse\n",
      "8604\t7 * 1229\tFalse\n",
      "8606\t5 * 1721\tFalse\n",
      "8608\t3 * 19 * 151\tFalse\n",
      "8610\t8609\tTrue\n",
      "8612\t79 * 109\tTrue\n",
      "8614\t3^3 * 11 * 29\tFalse\n",
      "8616\t5 * 1723\tFalse\n",
      "8618\t7 * 1231\tFalse\n",
      "8620\t3 * 13^2 * 17\tFalse\n",
      "8622\t37 * 233\tTrue\n",
      "8624\t8623\tFalse\n",
      "8626\t3 * 5^3 * 23\tFalse\n",
      "8628\t8627\tFalse\n",
      "8630\t8629\tTrue\n",
      "8632\t3^2 * 7 * 137\tFalse\n",
      "8634\t89 * 97\tFalse\n",
      "8636\t5 * 11 * 157\tFalse\n",
      "8638\t3 * 2879\tFalse\n",
      "8640\t53 * 163\tTrue\n",
      "8642\t8641\tTrue\n",
      "8644\t3 * 43 * 67\tFalse\n",
      "8646\t5 * 7 * 13 * 19\tFalse\n",
      "8648\t8647\tFalse\n",
      "8650\t3^2 * 31^2\tFalse\n",
      "8652\t41 * 211\tTrue\n",
      "8654\t17 * 509\tFalse\n",
      "8656\t3 * 5 * 577\tFalse\n",
      "8658\t11 * 787\tFalse\n",
      "8660\t7 * 1237\tTrue\n",
      "8662\t3 * 2887\tFalse\n",
      "8664\t8663\tFalse\n",
      "8666\t5 * 1733\tFalse\n",
      "8668\t3^4 * 107\tFalse\n",
      "8670\t8669\tTrue\n",
      "8672\t13 * 23 * 29\tTrue\n",
      "8674\t3 * 7^2 * 59\tFalse\n",
      "8676\t5^2 * 347\tFalse\n",
      "8678\t8677\tFalse\n",
      "8680\t3 * 11 * 263\tTrue\n",
      "8682\t8681\tTrue\n",
      "8684\t19 * 457\tFalse\n",
      "8686\t3^2 * 5 * 193\tFalse\n",
      "8688\t7 * 17 * 73\tFalse\n",
      "8690\t8689\tTrue\n",
      "8692\t3 * 2897\tFalse\n",
      "8694\t8693\tFalse\n",
      "8696\t5 * 37 * 47\tFalse\n",
      "8698\t3 * 13 * 223\tFalse\n",
      "8700\t8699\tTrue\n",
      "8702\t7 * 11 * 113\tTrue\n",
      "8704\t3^2 * 967\tFalse\n",
      "8706\t5 * 1741\tFalse\n",
      "8708\t8707\tFalse\n",
      "8710\t3 * 2903\tFalse\n",
      "8712\t31 * 281\tTrue\n",
      "8714\t8713\tFalse\n",
      "8716\t3 * 5 * 7 * 83\tFalse\n",
      "8718\t23 * 379\tFalse\n",
      "8720\t8719\tTrue\n",
      "8722\t3^3 * 17 * 19\tTrue\n",
      "8724\t11 * 13 * 61\tFalse\n",
      "8726\t5^2 * 349\tFalse\n",
      "8728\t3 * 2909\tFalse\n",
      "8730\t7 * 29 * 43\tTrue\n",
      "8732\t8731\tTrue\n",
      "8734\t3 * 41 * 71\tFalse\n",
      "8736\t5 * 1747\tFalse\n",
      "8738\t8737\tFalse\n",
      "8740\t3^2 * 971\tFalse\n",
      "8742\t8741\tTrue\n",
      "8744\t7 * 1249\tFalse\n",
      "8746\t3 * 5 * 11 * 53\tFalse\n",
      "8748\t8747\tFalse\n",
      "8750\t13 * 673\tTrue\n",
      "8752\t3 * 2917\tFalse\n",
      "8754\t8753\tFalse\n",
      "8756\t5 * 17 * 103\tFalse\n",
      "8758\t3^2 * 7 * 139\tFalse\n",
      "8760\t19 * 461\tTrue\n",
      "8762\t8761\tTrue\n",
      "8764\t3 * 23 * 127\tFalse\n",
      "8766\t5 * 1753\tFalse\n",
      "8768\t11 * 797\tFalse\n",
      "8770\t3 * 37 * 79\tTrue\n",
      "8772\t7^2 * 179\tTrue\n",
      "8774\t31 * 283\tFalse\n",
      "8776\t3^3 * 5^2 * 13\tFalse\n",
      "8778\t67 * 131\tFalse\n",
      "8780\t8779\tTrue\n",
      "8782\t3 * 2927\tFalse\n",
      "8784\t8783\tFalse\n",
      "8786\t5 * 7 * 251\tFalse\n",
      "8788\t3 * 29 * 101\tFalse\n",
      "8790\t11 * 17 * 47\tTrue\n",
      "8792\t59 * 149\tTrue\n",
      "8794\t3^2 * 977\tFalse\n",
      "8796\t5 * 1759\tFalse\n",
      "8798\t19 * 463\tFalse\n",
      "8800\t3 * 7 * 419\tFalse\n",
      "8802\t13 * 677\tTrue\n",
      "8804\t8803\tFalse\n",
      "8806\t3 * 5 * 587\tFalse\n",
      "8808\t8807\tFalse\n",
      "8810\t23 * 383\tTrue\n",
      "8812\t3^2 * 11 * 89\tTrue\n",
      "8814\t7 * 1259\tFalse\n",
      "8816\t5 * 41 * 43\tFalse\n",
      "8818\t3 * 2939\tFalse\n",
      "8820\t8819\tTrue\n",
      "8822\t8821\tTrue\n",
      "8824\t3 * 17 * 173\tFalse\n",
      "8826\t5^2 * 353\tFalse\n",
      "8828\t7 * 13 * 97\tFalse\n",
      "8830\t3^4 * 109\tFalse\n",
      "8832\t8831\tTrue\n",
      "8834\t11^2 * 73\tFalse\n",
      "8836\t3 * 5 * 19 * 31\tFalse\n",
      "8838\t8837\tFalse\n",
      "8840\t8839\tTrue\n",
      "8842\t3 * 7 * 421\tFalse\n",
      "8844\t37 * 239\tFalse\n",
      "8846\t5 * 29 * 61\tFalse\n",
      "8848\t3^2 * 983\tFalse\n",
      "8850\t8849\tTrue\n",
      "8852\t53 * 167\tTrue\n",
      "8854\t3 * 13 * 227\tFalse\n",
      "8856\t5 * 7 * 11 * 23\tFalse\n",
      "8858\t17 * 521\tFalse\n",
      "8860\t3 * 2953\tFalse\n",
      "8862\t8861\tTrue\n",
      "8864\t8863\tFalse\n",
      "8866\t3^2 * 5 * 197\tFalse\n",
      "8868\t8867\tFalse\n",
      "8870\t7^2 * 181\tTrue\n",
      "8872\t3 * 2957\tFalse\n",
      "8874\t19 * 467\tFalse\n",
      "8876\t5^3 * 71\tFalse\n",
      "8878\t3 * 11 * 269\tFalse\n",
      "8880\t13 * 683\tTrue\n",
      "8882\t83 * 107\tTrue\n",
      "8884\t3^3 * 7 * 47\tFalse\n",
      "8886\t5 * 1777\tFalse\n",
      "8888\t8887\tFalse\n",
      "8890\t3 * 2963\tFalse\n",
      "8892\t17 * 523\tTrue\n",
      "8894\t8893\tFalse\n",
      "8896\t3 * 5 * 593\tFalse\n",
      "8898\t7 * 31 * 41\tFalse\n",
      "8900\t11 * 809\tTrue\n",
      "8902\t3^2 * 23 * 43\tFalse\n",
      "8904\t29 * 307\tFalse\n",
      "8906\t5 * 13 * 137\tFalse\n",
      "8908\t3 * 2969\tFalse\n",
      "8910\t59 * 151\tTrue\n",
      "8912\t7 * 19 * 67\tTrue\n",
      "8914\t3 * 2971\tFalse\n",
      "8916\t5 * 1783\tFalse\n",
      "8918\t37 * 241\tFalse\n",
      "8920\t3^2 * 991\tFalse\n",
      "8922\t11 * 811\tTrue\n",
      "8924\t8923\tFalse\n",
      "8926\t3 * 5^2 * 7 * 17\tFalse\n",
      "8928\t79 * 113\tFalse\n",
      "8930\t8929\tTrue\n",
      "8932\t3 * 13 * 229\tFalse\n",
      "8934\t8933\tFalse\n",
      "8936\t5 * 1787\tFalse\n",
      "8938\t3^3 * 331\tFalse\n",
      "8940\t7 * 1277\tTrue\n",
      "8942\t8941\tTrue\n",
      "8944\t3 * 11 * 271\tFalse\n",
      "8946\t5 * 1789\tFalse\n",
      "8948\t23 * 389\tFalse\n",
      "8950\t3 * 19 * 157\tTrue\n",
      "8952\t8951\tTrue\n",
      "8954\t7 * 1279\tFalse\n",
      "8956\t3^2 * 5 * 199\tFalse\n",
      "8958\t13^2 * 53\tFalse\n",
      "8960\t17^2 * 31\tTrue\n",
      "8962\t3 * 29 * 103\tFalse\n",
      "8964\t8963\tFalse\n",
      "8966\t5 * 11 * 163\tFalse\n",
      "8968\t3 * 7^2 * 61\tFalse\n",
      "8970\t8969\tTrue\n",
      "8972\t8971\tTrue\n",
      "8974\t3^2 * 997\tFalse\n",
      "8976\t5^2 * 359\tFalse\n",
      "8978\t47 * 191\tFalse\n",
      "8980\t3 * 41 * 73\tFalse\n",
      "8982\t7 * 1283\tTrue\n",
      "8984\t13 * 691\tFalse\n",
      "8986\t3 * 5 * 599\tFalse\n",
      "8988\t11 * 19 * 43\tFalse\n",
      "8990\t89 * 101\tTrue\n",
      "8992\t3^5 * 37\tFalse\n",
      "8994\t17 * 23^2\tFalse\n",
      "8996\t5 * 7 * 257\tFalse\n",
      "8998\t3 * 2999\tFalse\n",
      "9000\t8999\tTrue\n",
      "9002\t9001\tTrue\n",
      "9004\t3 * 3001\tFalse\n",
      "9006\t5 * 1801\tFalse\n",
      "9008\t9007\tFalse\n",
      "9010\t3^2 * 7 * 11 * 13\tTrue\n",
      "9012\t9011\tTrue\n",
      "9014\t9013\tFalse\n",
      "9016\t3 * 5 * 601\tFalse\n",
      "9018\t71 * 127\tFalse\n",
      "9020\t29 * 311\tTrue\n",
      "9022\t3 * 31 * 97\tTrue\n",
      "9024\t7 * 1289\tFalse\n",
      "9026\t5^2 * 19^2\tFalse\n",
      "9028\t3^2 * 17 * 59\tFalse\n",
      "9030\t9029\tTrue\n",
      "9032\t11 * 821\tTrue\n",
      "9034\t3 * 3011\tFalse\n",
      "9036\t5 * 13 * 139\tFalse\n",
      "9038\t7 * 1291\tFalse\n",
      "9040\t3 * 23 * 131\tFalse\n",
      "9042\t9041\tTrue\n",
      "9044\t9043\tFalse\n",
      "9046\t3^3 * 5 * 67\tFalse\n",
      "9048\t83 * 109\tFalse\n",
      "9050\t9049\tTrue\n",
      "9052\t3 * 7 * 431\tFalse\n",
      "9054\t11 * 823\tFalse\n",
      "9056\t5 * 1811\tFalse\n",
      "9058\t3 * 3019\tFalse\n",
      "9060\t9059\tTrue\n",
      "9062\t13 * 17 * 41\tTrue\n",
      "9064\t3^2 * 19 * 53\tFalse\n",
      "9066\t5 * 7^2 * 37\tFalse\n",
      "9068\t9067\tFalse\n",
      "9070\t3 * 3023\tFalse\n",
      "9072\t47 * 193\tTrue\n",
      "9074\t43 * 211\tFalse\n",
      "9076\t3 * 5^2 * 11^2\tFalse\n",
      "9078\t29 * 313\tFalse\n",
      "9080\t7 * 1297\tTrue\n",
      "9082\t3^2 * 1009\tFalse\n",
      "9084\t31 * 293\tFalse\n",
      "9086\t5 * 23 * 79\tFalse\n",
      "9088\t3 * 13 * 233\tFalse\n",
      "9090\t61 * 149\tTrue\n",
      "9092\t9091\tTrue\n",
      "9094\t3 * 7 * 433\tFalse\n",
      "9096\t5 * 17 * 107\tFalse\n",
      "9098\t11 * 827\tFalse\n",
      "9100\t3^3 * 337\tFalse\n",
      "9102\t19 * 479\tTrue\n",
      "9104\t9103\tFalse\n",
      "9106\t3 * 5 * 607\tFalse\n",
      "9108\t7 * 1301\tFalse\n",
      "9110\t9109\tTrue\n",
      "9112\t3 * 3037\tFalse\n",
      "9114\t13 * 701\tFalse\n",
      "9116\t5 * 1823\tFalse\n",
      "9118\t3^2 * 1013\tFalse\n",
      "9120\t11 * 829\tTrue\n",
      "9122\t7 * 1303\tTrue\n",
      "9124\t3 * 3041\tFalse\n",
      "9126\t5^3 * 73\tFalse\n",
      "9128\t9127\tFalse\n",
      "9130\t3 * 17 * 179\tTrue\n",
      "9132\t23 * 397\tTrue\n",
      "9134\t9133\tFalse\n",
      "9136\t3^2 * 5 * 7 * 29\tFalse\n",
      "9138\t9137\tFalse\n",
      "9140\t13 * 19 * 37\tTrue\n",
      "9142\t3 * 11 * 277\tTrue\n",
      "9144\t41 * 223\tFalse\n",
      "9146\t5 * 31 * 59\tFalse\n",
      "9148\t3 * 3049\tFalse\n",
      "9150\t7 * 1307\tTrue\n",
      "9152\t9151\tTrue\n",
      "9154\t3^4 * 113\tFalse\n",
      "9156\t5 * 1831\tFalse\n",
      "9158\t9157\tFalse\n",
      "9160\t3 * 43 * 71\tFalse\n",
      "9162\t9161\tTrue\n",
      "9164\t7^2 * 11 * 17\tFalse\n",
      "9166\t3 * 5 * 13 * 47\tFalse\n",
      "9168\t89 * 103\tFalse\n",
      "9170\t53 * 173\tTrue\n",
      "9172\t3^2 * 1019\tFalse\n",
      "9174\t9173\tFalse\n",
      "9176\t5^2 * 367\tFalse\n",
      "9178\t3 * 7 * 19 * 23\tFalse\n",
      "9180\t67 * 137\tTrue\n",
      "9182\t9181\tTrue\n",
      "9184\t3 * 3061\tFalse\n",
      "9186\t5 * 11 * 167\tFalse\n",
      "9188\t9187\tFalse\n",
      "9190\t3^2 * 1021\tFalse\n",
      "9192\t7 * 13 * 101\tTrue\n",
      "9194\t29 * 317\tFalse\n",
      "9196\t3 * 5 * 613\tFalse\n",
      "9198\t17 * 541\tFalse\n",
      "9200\t9199\tTrue\n",
      "9202\t3 * 3067\tFalse\n",
      "9204\t9203\tFalse\n",
      "9206\t5 * 7 * 263\tFalse\n",
      "9208\t3^3 * 11 * 31\tFalse\n",
      "9210\t9209\tTrue\n",
      "9212\t61 * 151\tTrue\n",
      "9214\t3 * 37 * 83\tFalse\n",
      "9216\t5 * 19 * 97\tFalse\n",
      "9218\t13 * 709\tFalse\n",
      "9220\t3 * 7 * 439\tFalse\n",
      "9222\t9221\tTrue\n",
      "9224\t23 * 401\tFalse\n",
      "9226\t3^2 * 5^2 * 41\tFalse\n",
      "9228\t9227\tFalse\n",
      "9230\t11 * 839\tTrue\n",
      "9232\t3 * 17 * 181\tFalse\n",
      "9234\t7 * 1319\tFalse\n",
      "9236\t5 * 1847\tFalse\n",
      "9238\t3 * 3079\tFalse\n",
      "9240\t9239\tTrue\n",
      "9242\t9241\tTrue\n",
      "9244\t3^2 * 13 * 79\tFalse\n",
      "9246\t5 * 43^2\tFalse\n",
      "9248\t7 * 1321\tFalse\n",
      "9250\t3 * 3083\tFalse\n",
      "9252\t11 * 29^2\tTrue\n",
      "9254\t19 * 487\tFalse\n",
      "9256\t3 * 5 * 617\tFalse\n",
      "9258\t9257\tFalse\n",
      "9260\t47 * 197\tTrue\n",
      "9262\t3^3 * 7^3\tFalse\n",
      "9264\t59 * 157\tFalse\n",
      "9266\t5 * 17 * 109\tFalse\n",
      "9268\t3 * 3089\tFalse\n",
      "9270\t13 * 23 * 31\tTrue\n",
      "9272\t73 * 127\tTrue\n",
      "9274\t3 * 11 * 281\tFalse\n",
      "9276\t5^2 * 7 * 53\tFalse\n",
      "9278\t9277\tFalse\n",
      "9280\t3^2 * 1031\tFalse\n",
      "9282\t9281\tTrue\n",
      "9284\t9283\tFalse\n",
      "9286\t3 * 5 * 619\tFalse\n",
      "9288\t37 * 251\tFalse\n",
      "9290\t7 * 1327\tTrue\n",
      "9292\t3 * 19 * 163\tFalse\n",
      "9294\t9293\tFalse\n",
      "9296\t5 * 11 * 13^2\tFalse\n",
      "9298\t3^2 * 1033\tFalse\n",
      "9300\t17 * 547\tTrue\n",
      "9302\t71 * 131\tTrue\n",
      "9304\t3 * 7 * 443\tFalse\n",
      "9306\t5 * 1861\tFalse\n",
      "9308\t41 * 227\tFalse\n",
      "9310\t3 * 29 * 107\tFalse\n",
      "9312\t9311\tTrue\n",
      "9314\t67 * 139\tFalse\n",
      "9316\t3^4 * 5 * 23\tFalse\n",
      "9318\t7 * 11^3\tFalse\n",
      "9320\t9319\tTrue\n",
      "9322\t3 * 13 * 239\tTrue\n",
      "9324\t9323\tFalse\n",
      "9326\t5^2 * 373\tFalse\n",
      "9328\t3 * 3109\tFalse\n",
      "9330\t19 * 491\tTrue\n",
      "9332\t7 * 31 * 43\tTrue\n",
      "9334\t3^2 * 17 * 61\tFalse\n",
      "9336\t5 * 1867\tFalse\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9338\t9337\tFalse\n",
      "9340\t3 * 11 * 283\tTrue\n",
      "9342\t9341\tTrue\n",
      "9344\t9343\tFalse\n",
      "9346\t3 * 5 * 7 * 89\tFalse\n",
      "9348\t13 * 719\tFalse\n",
      "9350\t9349\tTrue\n",
      "9352\t3^2 * 1039\tFalse\n",
      "9354\t47 * 199\tFalse\n",
      "9356\t5 * 1871\tFalse\n",
      "9358\t3 * 3119\tFalse\n",
      "9360\t7^2 * 191\tTrue\n",
      "9362\t11 * 23 * 37\tTrue\n",
      "9364\t3 * 3121\tFalse\n",
      "9366\t5 * 1873\tFalse\n",
      "9368\t17 * 19 * 29\tFalse\n",
      "9370\t3^3 * 347\tFalse\n",
      "9372\t9371\tTrue\n",
      "9374\t7 * 13 * 103\tFalse\n",
      "9376\t3 * 5^5\tFalse\n",
      "9378\t9377\tFalse\n",
      "9380\t83 * 113\tTrue\n",
      "9382\t3 * 53 * 59\tTrue\n",
      "9384\t11 * 853\tFalse\n",
      "9386\t5 * 1877\tFalse\n",
      "9388\t3^2 * 7 * 149\tFalse\n",
      "9390\t41 * 229\tTrue\n",
      "9392\t9391\tTrue\n",
      "9394\t3 * 31 * 101\tFalse\n",
      "9396\t5 * 1879\tFalse\n",
      "9398\t9397\tFalse\n",
      "9400\t3 * 13 * 241\tTrue\n",
      "9402\t7 * 17 * 79\tTrue\n",
      "9404\t9403\tFalse\n",
      "9406\t3^2 * 5 * 11 * 19\tFalse\n",
      "9408\t23 * 409\tFalse\n",
      "9410\t97^2\tTrue\n",
      "9412\t3 * 3137\tFalse\n",
      "9414\t9413\tFalse\n",
      "9416\t5 * 7 * 269\tFalse\n",
      "9418\t3 * 43 * 73\tFalse\n",
      "9420\t9419\tTrue\n",
      "9422\t9421\tTrue\n",
      "9424\t3^3 * 349\tFalse\n",
      "9426\t5^2 * 13 * 29\tFalse\n",
      "9428\t11 * 857\tFalse\n",
      "9430\t3 * 7 * 449\tFalse\n",
      "9432\t9431\tTrue\n",
      "9434\t9433\tFalse\n",
      "9436\t3 * 5 * 17 * 37\tFalse\n",
      "9438\t9437\tFalse\n",
      "9440\t9439\tTrue\n",
      "9442\t3^2 * 1049\tFalse\n",
      "9444\t7 * 19 * 71\tFalse\n",
      "9446\t5 * 1889\tFalse\n",
      "9448\t3 * 47 * 67\tFalse\n",
      "9450\t11 * 859\tTrue\n",
      "9452\t13 * 727\tTrue\n",
      "9454\t3 * 23 * 137\tFalse\n",
      "9456\t5 * 31 * 61\tFalse\n",
      "9458\t7^2 * 193\tFalse\n",
      "9460\t3^2 * 1051\tFalse\n",
      "9462\t9461\tTrue\n",
      "9464\t9463\tFalse\n",
      "9466\t3 * 5 * 631\tFalse\n",
      "9468\t9467\tFalse\n",
      "9470\t17 * 557\tTrue\n",
      "9472\t3 * 7 * 11 * 41\tTrue\n",
      "9474\t9473\tFalse\n",
      "9476\t5^2 * 379\tFalse\n",
      "9478\t3^6 * 13\tFalse\n",
      "9480\t9479\tTrue\n",
      "9482\t19 * 499\tTrue\n",
      "9484\t3 * 29 * 109\tFalse\n",
      "9486\t5 * 7 * 271\tFalse\n",
      "9488\t53 * 179\tFalse\n",
      "9490\t3 * 3163\tFalse\n",
      "9492\t9491\tTrue\n",
      "9494\t11 * 863\tFalse\n",
      "9496\t3^2 * 5 * 211\tFalse\n",
      "9498\t9497\tFalse\n",
      "9500\t7 * 23 * 59\tTrue\n",
      "9502\t3 * 3167\tFalse\n",
      "9504\t13 * 17 * 43\tFalse\n",
      "9506\t5 * 1901\tFalse\n",
      "9508\t3 * 3169\tFalse\n",
      "9510\t37 * 257\tTrue\n",
      "9512\t9511\tTrue\n",
      "9514\t3^2 * 7 * 151\tFalse\n",
      "9516\t5 * 11 * 173\tFalse\n",
      "9518\t31 * 307\tFalse\n",
      "9520\t3 * 19 * 167\tFalse\n",
      "9522\t9521\tTrue\n",
      "9524\t89 * 107\tFalse\n",
      "9526\t3 * 5^2 * 127\tFalse\n",
      "9528\t7 * 1361\tFalse\n",
      "9530\t13 * 733\tTrue\n",
      "9532\t3^3 * 353\tFalse\n",
      "9534\t9533\tFalse\n",
      "9536\t5 * 1907\tFalse\n",
      "9538\t3 * 11 * 17^2\tFalse\n",
      "9540\t9539\tTrue\n",
      "9542\t7 * 29 * 47\tTrue\n",
      "9544\t3 * 3181\tFalse\n",
      "9546\t5 * 23 * 83\tFalse\n",
      "9548\t9547\tFalse\n",
      "9550\t3^2 * 1061\tFalse\n",
      "9552\t9551\tTrue\n",
      "9554\t41 * 233\tFalse\n",
      "9556\t3 * 5 * 7^2 * 13\tFalse\n",
      "9558\t19 * 503\tFalse\n",
      "9560\t11^2 * 79\tTrue\n",
      "9562\t3 * 3187\tFalse\n",
      "9564\t73 * 131\tFalse\n",
      "9566\t5 * 1913\tFalse\n",
      "9568\t3^2 * 1063\tFalse\n",
      "9570\t7 * 1367\tTrue\n",
      "9572\t17 * 563\tTrue\n",
      "9574\t3 * 3191\tFalse\n",
      "9576\t5^2 * 383\tFalse\n",
      "9578\t61 * 157\tFalse\n",
      "9580\t3 * 31 * 103\tFalse\n",
      "9582\t11 * 13 * 67\tTrue\n",
      "9584\t7 * 37^2\tFalse\n",
      "9586\t3^3 * 5 * 71\tFalse\n",
      "9588\t9587\tFalse\n",
      "9590\t43 * 223\tTrue\n",
      "9592\t3 * 23 * 139\tFalse\n",
      "9594\t53 * 181\tFalse\n",
      "9596\t5 * 19 * 101\tFalse\n",
      "9598\t3 * 7 * 457\tFalse\n",
      "9600\t29 * 331\tTrue\n",
      "9602\t9601\tTrue\n",
      "9604\t3^2 * 11 * 97\tFalse\n",
      "9606\t5 * 17 * 113\tFalse\n",
      "9608\t13 * 739\tFalse\n",
      "9610\t3 * 3203\tFalse\n",
      "9612\t7 * 1373\tTrue\n",
      "9614\t9613\tFalse\n",
      "9616\t3 * 5 * 641\tFalse\n",
      "9618\t59 * 163\tFalse\n",
      "9620\t9619\tTrue\n",
      "9622\t3^2 * 1069\tFalse\n",
      "9624\t9623\tFalse\n",
      "9626\t5^3 * 7 * 11\tFalse\n",
      "9628\t3 * 3209\tFalse\n",
      "9630\t9629\tTrue\n",
      "9632\t9631\tTrue\n",
      "9634\t3 * 13^2 * 19\tFalse\n",
      "9636\t5 * 41 * 47\tFalse\n",
      "9638\t23 * 419\tFalse\n",
      "9640\t3^4 * 7 * 17\tTrue\n",
      "9642\t31 * 311\tTrue\n",
      "9644\t9643\tFalse\n",
      "9646\t3 * 5 * 643\tFalse\n",
      "9648\t11 * 877\tFalse\n",
      "9650\t9649\tTrue\n",
      "9652\t3 * 3217\tFalse\n",
      "9654\t7^2 * 197\tFalse\n",
      "9656\t5 * 1931\tFalse\n",
      "9658\t3^2 * 29 * 37\tFalse\n",
      "9660\t13 * 743\tTrue\n",
      "9662\t9661\tTrue\n",
      "9664\t3 * 3221\tFalse\n",
      "9666\t5 * 1933\tFalse\n",
      "9668\t7 * 1381\tFalse\n",
      "9670\t3 * 11 * 293\tTrue\n",
      "9672\t19 * 509\tTrue\n",
      "9674\t17 * 569\tFalse\n",
      "9676\t3^2 * 5^2 * 43\tFalse\n",
      "9678\t9677\tFalse\n",
      "9680\t9679\tTrue\n",
      "9682\t3 * 7 * 461\tFalse\n",
      "9684\t23 * 421\tFalse\n",
      "9686\t5 * 13 * 149\tFalse\n",
      "9688\t3 * 3229\tFalse\n",
      "9690\t9689\tTrue\n",
      "9692\t11 * 881\tTrue\n",
      "9694\t3^3 * 359\tFalse\n",
      "9696\t5 * 7 * 277\tFalse\n",
      "9698\t9697\tFalse\n",
      "9700\t3 * 53 * 61\tFalse\n",
      "9702\t89 * 109\tTrue\n",
      "9704\t31 * 313\tFalse\n",
      "9706\t3 * 5 * 647\tFalse\n",
      "9708\t17 * 571\tFalse\n",
      "9710\t7 * 19 * 73\tTrue\n",
      "9712\t3^2 * 13 * 83\tTrue\n",
      "9714\t11 * 883\tFalse\n",
      "9716\t5 * 29 * 67\tFalse\n",
      "9718\t3 * 41 * 79\tFalse\n",
      "9720\t9719\tTrue\n",
      "9722\t9721\tTrue\n",
      "9724\t3 * 7 * 463\tFalse\n",
      "9726\t5^2 * 389\tFalse\n",
      "9728\t71 * 137\tFalse\n",
      "9730\t3^2 * 23 * 47\tFalse\n",
      "9732\t37 * 263\tTrue\n",
      "9734\t9733\tFalse\n",
      "9736\t3 * 5 * 11 * 59\tFalse\n",
      "9738\t7 * 13 * 107\tFalse\n",
      "9740\t9739\tTrue\n",
      "9742\t3 * 17 * 191\tTrue\n",
      "9744\t9743\tFalse\n",
      "9746\t5 * 1949\tFalse\n",
      "9748\t3^3 * 19^2\tFalse\n",
      "9750\t9749\tTrue\n",
      "9752\t7^2 * 199\tTrue\n",
      "9754\t3 * 3251\tFalse\n",
      "9756\t5 * 1951\tFalse\n",
      "9758\t11 * 887\tFalse\n",
      "9760\t3 * 3253\tFalse\n",
      "9762\t43 * 227\tTrue\n",
      "9764\t13 * 751\tFalse\n",
      "9766\t3^2 * 5 * 7 * 31\tFalse\n",
      "9768\t9767\tFalse\n",
      "9770\t9769\tTrue\n",
      "9772\t3 * 3257\tFalse\n",
      "9774\t29 * 337\tFalse\n",
      "9776\t5^2 * 17 * 23\tFalse\n",
      "9778\t3 * 3259\tFalse\n",
      "9780\t7 * 11 * 127\tTrue\n",
      "9782\t9781\tTrue\n",
      "9784\t3^2 * 1087\tFalse\n",
      "9786\t5 * 19 * 103\tFalse\n",
      "9788\t9787\tFalse\n",
      "9790\t3 * 13 * 251\tTrue\n",
      "9792\t9791\tTrue\n",
      "9794\t7 * 1399\tFalse\n",
      "9796\t3 * 5 * 653\tFalse\n",
      "9798\t97 * 101\tFalse\n",
      "9800\t41 * 239\tTrue\n",
      "9802\t3^4 * 11^2\tFalse\n",
      "9804\t9803\tFalse\n",
      "9806\t5 * 37 * 53\tFalse\n",
      "9808\t3 * 7 * 467\tFalse\n",
      "9810\t17 * 577\tTrue\n",
      "9812\t9811\tTrue\n",
      "9814\t3 * 3271\tFalse\n",
      "9816\t5 * 13 * 151\tFalse\n",
      "9818\t9817\tFalse\n",
      "9820\t3^2 * 1091\tFalse\n",
      "9822\t7 * 23 * 61\tTrue\n",
      "9824\t11 * 19 * 47\tFalse\n",
      "9826\t3 * 5^2 * 131\tFalse\n",
      "9828\t31 * 317\tFalse\n",
      "9830\t9829\tTrue\n",
      "9832\t3 * 29 * 113\tFalse\n",
      "9834\t9833\tFalse\n",
      "9836\t5 * 7 * 281\tFalse\n",
      "9838\t3^2 * 1093\tFalse\n",
      "9840\t9839\tTrue\n",
      "9842\t13 * 757\tTrue\n",
      "9844\t3 * 17 * 193\tFalse\n",
      "9846\t5 * 11 * 179\tFalse\n",
      "9848\t43 * 229\tFalse\n",
      "9850\t3 * 7^2 * 67\tFalse\n",
      "9852\t9851\tTrue\n",
      "9854\t59 * 167\tFalse\n",
      "9856\t3^3 * 5 * 73\tFalse\n",
      "9858\t9857\tFalse\n",
      "9860\t9859\tTrue\n",
      "9862\t3 * 19 * 173\tTrue\n",
      "9864\t7 * 1409\tFalse\n",
      "9866\t5 * 1973\tFalse\n",
      "9868\t3 * 11 * 13 * 23\tFalse\n",
      "9870\t71 * 139\tTrue\n",
      "9872\t9871\tTrue\n",
      "9874\t3^2 * 1097\tFalse\n",
      "9876\t5^3 * 79\tFalse\n",
      "9878\t7 * 17 * 83\tFalse\n",
      "9880\t3 * 37 * 89\tFalse\n",
      "9882\t41 * 241\tTrue\n",
      "9884\t9883\tFalse\n",
      "9886\t3 * 5 * 659\tFalse\n",
      "9888\t9887\tFalse\n",
      "9890\t11 * 29 * 31\tTrue\n",
      "9892\t3^2 * 7 * 157\tFalse\n",
      "9894\t13 * 761\tFalse\n",
      "9896\t5 * 1979\tFalse\n",
      "9898\t3 * 3299\tFalse\n",
      "9900\t19 * 521\tTrue\n",
      "9902\t9901\tTrue\n",
      "9904\t3 * 3301\tFalse\n",
      "9906\t5 * 7 * 283\tFalse\n",
      "9908\t9907\tFalse\n",
      "9910\t3^3 * 367\tFalse\n",
      "9912\t11 * 17 * 53\tTrue\n",
      "9914\t23 * 431\tFalse\n",
      "9916\t3 * 5 * 661\tFalse\n",
      "9918\t47 * 211\tFalse\n",
      "9920\t7 * 13 * 109\tTrue\n",
      "9922\t3 * 3307\tFalse\n",
      "9924\t9923\tFalse\n",
      "9926\t5^2 * 397\tFalse\n",
      "9928\t3^2 * 1103\tFalse\n",
      "9930\t9929\tTrue\n",
      "9932\t9931\tTrue\n",
      "9934\t3 * 7 * 11 * 43\tFalse\n",
      "9936\t5 * 1987\tFalse\n",
      "9938\t19 * 523\tFalse\n",
      "9940\t3 * 3313\tFalse\n",
      "9942\t9941\tTrue\n",
      "9944\t61 * 163\tFalse\n",
      "9946\t3^2 * 5 * 13 * 17\tFalse\n",
      "9948\t7^3 * 29\tFalse\n",
      "9950\t9949\tTrue\n",
      "9952\t3 * 31 * 107\tFalse\n",
      "9954\t37 * 269\tFalse\n",
      "9956\t5 * 11 * 181\tFalse\n",
      "9958\t3 * 3319\tFalse\n",
      "9960\t23 * 433\tTrue\n",
      "9962\t7 * 1423\tTrue\n",
      "9964\t3^5 * 41\tFalse\n",
      "9966\t5 * 1993\tFalse\n",
      "9968\t9967\tFalse\n",
      "9970\t3 * 3323\tFalse\n",
      "9972\t13^2 * 59\tTrue\n",
      "9974\t9973\tFalse\n",
      "9976\t3 * 5^2 * 7 * 19\tFalse\n",
      "9978\t11 * 907\tFalse\n",
      "9980\t17 * 587\tTrue\n",
      "9982\t3^2 * 1109\tFalse\n",
      "9984\t67 * 149\tFalse\n",
      "9986\t5 * 1997\tFalse\n",
      "9988\t3 * 3329\tFalse\n",
      "9990\t7 * 1427\tTrue\n",
      "9992\t97 * 103\tTrue\n",
      "9994\t3 * 3331\tFalse\n",
      "9996\t5 * 1999\tFalse\n",
      "9998\t13 * 769\tFalse\n",
      "10000\t3^2 * 11 * 101\tTrue\n",
      "{1}\n",
      "{3081, 7689, 9741, 8721, 7701, 6681, 5661, 2079, 4641, 8229, 3621, 6699, 1581, 561, 6201, 9789, 2109, 5181, 8769, 4161, 5709, 7761, 4191, 9321, 8811, 9339, 3201, 7809, 9861, 6789, 7821, 3729, 7839, 2211, 9381, 6831, 2739, 5811, 5301, 9399, 1209, 8379, 7359, 1221, 7881, 2769, 5841, 1749, 6369, 741, 4329, 4851, 8949, 759, 2301, 9471, 5889, 5379, 9999, 3861, 7449, 8481, 4389, 6441, 5421, 9009, 2871, 9021, 4929, 7491, 6981, 3399, 8019, 1881, 8541, 3939, 6501, 2409, 3441, 7029, 1911, 891, 5511, 1419, 6549, 6039, 5529, 9639, 5031, 9129, 4521, 429, 8109, 7089, 6579, 9141, 4029, 3009, 9669, 1989, 969, 3531, 3021, 4059, 8679, 9711, 5109, 4599, 7161, 3069}\n"
     ]
    }
   ],
   "source": [
    "#予想''の検証\n",
    "N=10000\n",
    "Q = []\n",
    "R = []\n",
    "for deg in range(2,N+1,2):\n",
    "    q = deg-1\n",
    "    if((q%10==1 or q%10==9) and q%2!=0 and q%3!=0 and q%5!=0):\n",
    "        Q.append(q)\n",
    "    if(dim(deg//2)==0):\n",
    "        continue\n",
    "    \n",
    "    #if(is_prime_power(deg-1)):\n",
    "    #    p,k = factor(deg-1)[0]\n",
    "    #else:\n",
    "    #    continue\n",
    "    ok = True\n",
    "    for (p,k) in factor(q):\n",
    "        Ord = [0 for i in range(deg+1)] #何回割れるか\n",
    "        for m in range(1,k+1):\n",
    "            f = awesome_polynomial_mod(deg, p^m)\n",
    "            #show(f)\n",
    "            for i, c in enumerate(f.list()):\n",
    "                if(c == 0):\n",
    "                    Ord[i] = m;\n",
    "        #print(Ord)\n",
    "        ok = ok and all([ord_p(i,p)+ord_p(deg-i,p)+r >= k for i,r in enumerate(Ord)])\n",
    "    if(ok):\n",
    "        R.append(q)\n",
    "    print(f\"{deg}\\t{factor(q)}\\t{ok}\") \n",
    "    \n",
    "print(set(Q)-set(R))\n",
    "print(set(R)-set(Q))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 * 11 * 13\n",
      "3 * 11 * 17\n",
      "3 * 13 * 19\n",
      "3 * 11 * 23\n",
      "3^4 * 11\n",
      "3 * 17 * 19\n",
      "3 * 13 * 31\n",
      "3 * 11 * 37\n",
      "3 * 11 * 43\n",
      "3 * 17 * 31\n",
      "3 * 11 * 53\n",
      "3^2 * 11 * 19\n",
      "3 * 7^2 * 13\n",
      "3^2 * 13 * 17\n",
      "3^3 * 7 * 11\n",
      "3 * 19 * 37\n",
      "3 * 11 * 67\n",
      "3 * 13 * 59\n",
      "3 * 11 * 73\n",
      "3 * 11 * 83\n",
      "3 * 13 * 71\n",
      "3^2 * 11 * 29\n",
      "3 * 17 * 59\n",
      "3 * 19 * 53\n",
      "3^2 * 11 * 31\n",
      "3 * 13 * 79\n",
      "3 * 11 * 97\n",
      "3 * 11 * 103\n",
      "3 * 31 * 37\n",
      "3 * 11 * 107\n",
      "3 * 17 * 71\n",
      "3 * 11 * 113\n",
      "3^3 * 11 * 13\n",
      "3 * 13 * 101\n",
      "3 * 17 * 79\n",
      "3^2 * 11 * 41\n",
      "3 * 19 * 73\n",
      "3 * 11 * 127\n",
      "3^2 * 13 * 37\n",
      "3 * 7 * 11 * 19\n",
      "3 * 11 * 137\n",
      "3^2 * 7 * 73\n",
      "3 * 7 * 13 * 17\n",
      "3^2 * 7^2 * 11\n",
      "3 * 31 * 53\n",
      "3^2 * 13 * 43\n",
      "3 * 13 * 131\n",
      "3 * 11 * 157\n",
      "3^2 * 19 * 31\n",
      "3 * 11 * 163\n",
      "3 * 13 * 139\n",
      "3 * 11 * 167\n",
      "3 * 19 * 97\n",
      "3^2 * 17 * 37\n",
      "3 * 11 * 173\n",
      "3 * 13 * 149\n",
      "3^2 * 11 * 59\n",
      "3 * 13 * 151\n",
      "3^2 * 11 * 61\n",
      "3^2 * 13 * 53\n",
      "3 * 11 * 193\n",
      "3 * 19 * 113\n",
      "3 * 11 * 197\n",
      "3 * 37 * 59\n",
      "3^2 * 17 * 43\n",
      "3 * 17 * 131\n",
      "3 * 7 * 11 * 29\n",
      "3 * 31 * 73\n",
      "3^3 * 11 * 23\n",
      "3 * 13 * 179\n",
      "3^2 * 11 * 71\n",
      "3 * 17 * 139\n",
      "3 * 7 * 11 * 31\n",
      "3 * 11 * 223\n",
      "3 * 13 * 191\n",
      "3 * 11 * 227\n",
      "3 * 11 * 233\n",
      "3 * 17 * 151\n",
      "3 * 13 * 199\n",
      "3 * 19 * 137\n",
      "3^2 * 11 * 79\n",
      "3^2 * 13 * 67\n",
      "3 * 37 * 71\n",
      "3^6 * 11\n",
      "3^2 * 17 * 53\n",
      "3 * 13 * 211\n",
      "3^2 * 7^2 * 19\n",
      "3 * 11 * 257\n",
      "3^2 * 13 * 73\n",
      "3 * 11 * 263\n",
      "3^3 * 17 * 19\n",
      "3 * 37 * 79\n",
      "3^2 * 11 * 89\n",
      "3 * 19 * 157\n",
      "3^2 * 7 * 11 * 13\n",
      "3 * 31 * 97\n",
      "3 * 17 * 179\n",
      "3 * 11 * 277\n",
      "3 * 13 * 239\n",
      "3 * 11 * 283\n",
      "3 * 53 * 59\n",
      "3 * 13 * 241\n",
      "3 * 7 * 11 * 41\n",
      "3^4 * 7 * 17\n",
      "3 * 11 * 293\n",
      "3^2 * 13 * 83\n",
      "3 * 17 * 191\n",
      "3 * 13 * 251\n",
      "3 * 19 * 173\n",
      "3^2 * 11 * 101\n"
     ]
    }
   ],
   "source": [
    "#例外的な次数たち\n",
    "E = sorted(list(set(R)-set(Q)))\n",
    "for n in E:\n",
    "    print(factor(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "429, 561, 741, 759, 891, 969, 1209, 1221, 1419, 1581, \\\\\n",
      "1749, 1881, 1911, 1989, 2079, 2109, 2211, 2301, 2409, 2739, \\\\\n",
      "2769, 2871, 3009, 3021, 3069, 3081, 3201, 3399, 3441, 3531, \\\\\n",
      "3621, 3729, 3861, 3939, 4029, 4059, 4161, 4191, 4329, 4389, \\\\\n",
      "4521, 4599, 4641, 4851, 4929, 5031, 5109, 5181, 5301, 5379, \\\\\n",
      "5421, 5511, 5529, 5661, 5709, 5811, 5841, 5889, 6039, 6201, \\\\\n",
      "6369, 6441, 6501, 6549, 6579, 6681, 6699, 6789, 6831, 6981, \\\\\n",
      "7029, 7089, 7161, 7359, 7449, 7491, 7689, 7701, 7761, 7809, \\\\\n",
      "7821, 7839, 7881, 8019, 8109, 8229, 8379, 8481, 8541, 8679, \\\\\n",
      "8721, 8769, 8811, 8949, 9009, 9021, 9129, 9141, 9321, 9339, \\\\\n",
      "9381, 9399, 9471, 9639, 9669, 9711, 9741, 9789, 9861, 9999, \\\\\n",
      "\n"
     ]
    }
   ],
   "source": [
    "s = \"\"\n",
    "for i,e in enumerate(E):\n",
    "    s += f\"{e}, \"\n",
    "    if(i%10==9):\n",
    "        s += \"\\\\\\\\\\n\"\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "110"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#アニメーション生成用\n",
    "\n",
    "#https://scipython.com/book/chapter-8-scipy/examples/visualizing-the-spherical-harmonics/\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "from matplotlib import cm, colors\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import numpy as np\n",
    "from numpy.core.records import array\n",
    "from scipy.special import sph_harm\n",
    "\n",
    "def superpose(coefficients, theta, phi):\n",
    "    val = 0\n",
    "    for t in coefficients:\n",
    "        l,m,c = t\n",
    "        val += c*sph_harm(m,l,theta,phi)\n",
    "    return val\n",
    "\n",
    "fig = plt.figure(figsize=plt.figaspect(1.), facecolor='none')\n",
    "fig.subplots_adjust(left=0, right=1, bottom=0, top=1)\n",
    "\n",
    "ax = fig.add_subplot(111, projection='3d', facecolor='none')\n",
    "ax.set_axis_off()\n",
    "ax.set_box_aspect((1,1,1)) #https://sabopy.com/py/matplotlib-3d-50/\n",
    "\n",
    "fcolors = np.array([])\n",
    "#resolution=dict(phi=100, theta=100)\n",
    "\n",
    "def equal_area_phi(n):\n",
    "    A = np.cumsum(1/np.sin(np.arange(1,n+1)*np.pi/(n+1)))\n",
    "    return A * np.pi / A[-1]\n",
    "\n",
    "def plot_spherical_harmonics(ax, coefficients, deg, elev=0.0, azim=0.0, is_first=True):\n",
    "    global fcolors\n",
    "    #global resolution\n",
    "    phi = np.linspace(0, np.pi, 180)\n",
    "    theta = np.linspace(0, 2*np.pi, 360)\n",
    "    phi, theta = np.meshgrid(phi, theta)\n",
    "\n",
    "    x = np.sin(phi) * np.cos(theta)\n",
    "    y = np.sin(phi) * np.sin(theta)\n",
    "    z = np.cos(phi)\n",
    "\n",
    "    if(is_first):\n",
    "        if(deg%4==0):\n",
    "            fcolors = superpose(coefficients, theta, phi).real\n",
    "        elif(deg%4==2):\n",
    "            fcolors = superpose(coefficients, theta, phi).imag\n",
    "        \n",
    "    fmax, fmin = fcolors.max(), fcolors.min()\n",
    "    #fcolors = (fcolors - fmin)/(fmax - fmin)\n",
    "    norm = colors.Normalize(vmin=fmin, vmax=fmax)\n",
    "\n",
    "    ax.view_init(elev=elev, azim=azim)\n",
    "    ax.plot_surface(x, y, z,  rstride=1, cstride=1, facecolors=plt.cm.jet(norm(fcolors)), shade=False)\n",
    "    plt.plot(1,1)\n",
    "\n",
    "interval = 75\n",
    "omega = 6\n",
    "frames = 12\n",
    "    \n",
    "def plot(i, deg, coefficients):\n",
    "    print(f\"{i+1}/{frames}...\")\n",
    "    plot_spherical_harmonics(ax, coefficients, deg, elev=30, azim=omega*i, is_first=True if i==0 else False)\n",
    "\n",
    "def animate(f, deg, file=\"out.gif\"):\n",
    "    v = get_normalized_coefficients(f, deg)\n",
    "    coefficients = []\n",
    "    for i, c in enumerate(v):\n",
    "        coefficients.append((deg//2, deg//2-i, c.n()))\n",
    "    ani = animation.FuncAnimation(fig, lambda i:plot(i,deg,coefficients), interval=interval, frames=frames)\n",
    "    #ani.repeat()\n",
    "    ani.save(file, writer=\"imagemagick\", savefig_kwargs=dict(facecolor=\"black\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X^71 + 194327*X^61 - 9577545*X^56 - 42805829*X^51 + 876828771*X^46 - 951230665*X^41 + 800021397*X^36 + 951230665*X^31 + 876828771*X^26 + 42805829*X^21 - 9577545*X^16 - 194327*X^11 - X"
      ]
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {},
   "outputs": [],
   "source": [
    "deg = 72\n",
    "f = (awesome_polynomial(deg))(x/y) * y^deg\n",
    "f = f.expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "interval = 150\n",
    "omega = 6\n",
    "frames = 12\n",
    "    \n",
    "animate(f, deg, file=\"rotating-icosahedron-deg-72-rep.gif\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.7"
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 },
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