From a23ba30bbd33d3cc5edfa374eab5d808561fe3a4 Mon Sep 17 00:00:00 2001
From: Jeremy Teitelbaum <jeremy.teitelbaum@uconn.edu>
Date: Thu, 5 Jul 2018 15:28:21 -0400
Subject: [PATCH 1/2] Delete Untitled1.ipynb

---
 ctnt2018/Untitled1.ipynb | 374 ---------------------------------------
 1 file changed, 374 deletions(-)
 delete mode 100644 ctnt2018/Untitled1.ipynb

diff --git a/ctnt2018/Untitled1.ipynb b/ctnt2018/Untitled1.ipynb
deleted file mode 100644
index 26bc131..0000000
--- a/ctnt2018/Untitled1.ipynb
+++ /dev/null
@@ -1,374 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "U=59649589127497217"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "N=2**(128)+1"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 4,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "V=N//U"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 5,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "P=np.zeros(10000)\n",
-    "P[0]=1\n",
-    "P[1]=1\n",
-    "for i in range(2,10000):\n",
-    "    if P[i]==0:\n",
-    "        j=2\n",
-    "        while i*j<10000:\n",
-    "            P[i*j]=1\n",
-    "            j=j+1\n",
-    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "K=U-1"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 8,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "2\n"
-     ]
-    }
-   ],
-   "source": [
-    "for x in Primes10000:\n",
-    "    if K % x ==0:\n",
-    "        print(x)\n",
-    "        K=K//x"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "def mexp(a,x,N):\n",
-    "    m,s=1,a\n",
-    "    while x>0:\n",
-    "        if x % 2 ==1:\n",
-    "            m=((m*s) % N)\n",
-    "        s=((s*s) % N)\n",
-    "        x=x//2\n",
-    "    return m"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "59649589127497216"
-      ]
-     },
-     "execution_count": 11,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "mexp(3,K,U)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 13,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# %load ECM.py\n",
-    "# %load ec3.py\n",
-    "from math import factorial, gcd, log\n",
-    "import numpy as np\n",
-    "P=np.zeros(1000)\n",
-    "P[0]=1\n",
-    "P[1]=1\n",
-    "for i in range(2,100):\n",
-    "    if P[i]==0:\n",
-    "        j=2\n",
-    "        while i*j<100:\n",
-    "            P[i*j]=1\n",
-    "            j=j+1\n",
-    "Primes1000=[i for i,x in enumerate(P) if x==0 ]  \n",
-    "\n",
-    "\n",
-    "def mexp(a,x,N):\n",
-    "    m,s=1,a\n",
-    "    while x>0:\n",
-    "        if x % 2 ==1:\n",
-    "            m=((m*s) % N)\n",
-    "        s=((s*s) % N)\n",
-    "        x=x//2\n",
-    "    return m\n",
-    "\n",
-    "def euclid(u,v):\n",
-    "    if v==0:\n",
-    "        raise ArithmeticError('Division by Zero')\n",
-    "    x0,x1=u,v\n",
-    "    a0,a1=1,0\n",
-    "    b0,b1=0,1\n",
-    "    while x1!=0:\n",
-    "        q=x0//x1\n",
-    "        x2=x0-q*x1\n",
-    "        a2=a0-q*a1\n",
-    "        b2=b0-q*b1\n",
-    "        x0,a0,b0=x1,a1,b1\n",
-    "        x1,a1,b1=x2,a2,b2\n",
-    "    if x0<0:\n",
-    "        return -x0,-a0,-b0\n",
-    "    else:\n",
-    "        return x0,a0,b0\n",
-    "\n",
-    "def mod_inv(u,N):\n",
-    "    d,a,b=euclid(u,N)\n",
-    "    if d==1:\n",
-    "        return a\n",
-    "    else:\n",
-    "        raise ArithmeticError('Common factor is '+str(d))\n",
-    "\n",
-    "def two_p(x,y,a,b,N):\n",
-    "    Lu=(3*x**2+a) % N\n",
-    "   # print(Lu)\n",
-    "    Lb=mod_inv(2*y,N)\n",
-    "  #  print(Lb)\n",
-    "    L=Lu*Lb % N\n",
-    "    x_two=(L*L-2*x) % N\n",
-    "    y_two=(L*(x-x_two)-y) %N\n",
-    "    return x_two,y_two\n",
-    "\n",
-    "def sum_p(x1,y1,x2,y2,a,b,N):\n",
-    "    Lu=(y2-y1) % N\n",
-    "    Lb=mod_inv(x2-x1,N)\n",
-    "    L=(Lu*Lb) % N\n",
-    "    x_sum=(L*L-x1-x2) %N\n",
-    "    y_sum=(L*(x1-x_sum)-y1) %N\n",
-    "    return x_sum,y_sum\n",
-    "    \n",
-    "def exp_p(x,y,a,b,m,N):\n",
-    "    sx,sy=x,y\n",
-    "    first=True\n",
-    "    while m>0:\n",
-    "        if m%2==1:\n",
-    "            if first:\n",
-    "                xm,ym=sx,sy\n",
-    "                first=False\n",
-    "            else:\n",
-    "                xm,ym=sum_p(xm,ym,sx,sy,a,b,N)\n",
-    "        sx,sy=two_p(sx,sy,a,b,N)\n",
-    "        m=m//2\n",
-    "    return xm,ym\n",
-    "\n",
-    "#def mexp(a,x,N):\n",
-    " #   m,s=1,a\n",
-    " #   while x>0:\n",
-    " #       if x % 2 ==1:\n",
-    " #           m=((m*s) % N)\n",
-    " #       s=((s*s) % N)\n",
-    " #       x=x//2\n",
-    " #   return m\n",
-    "\n",
-    "def ecm_trial(N,arange=50,krange=30):\n",
-    "    for a in range(-arange,arange):\n",
-    "        xm,ym=0,1\n",
-    "        print(a) \n",
-    "        for k in range(2,krange):\n",
-    "            try:\n",
-    "                xm,ym=exp_p(xm,ym,a,1,k,N)\n",
-    "            except ArithmeticError:\n",
-    "                print('try the following: a=',a,' and k=',k)\n",
-    "                break\n",
-    "\n",
-    "                        \n",
-    "                    \n",
-    "#N=149185656432189838133\n",
-    "#ecm_trial(N,arange=20,krange=10000)\n",
-    "N=2**128+1\n",
-    "#ecm_trial(N,arange=100,krange=10000)\n",
-    "#xm,ym=exp_p(0,1,-91,1,factorial(7883),N)\n",
-    "                                                                                                                                    \n",
-    "                                            \n",
-    "\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 19,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "-100\n",
-      "-99\n",
-      "-98\n",
-      "-97\n",
-      "-96\n",
-      "-95\n",
-      "-94\n",
-      "-93\n",
-      "-92\n",
-      "-91\n",
-      "try the following: a= -91  and k= 7883\n",
-      "-90\n",
-      "-89\n",
-      "-88\n",
-      "-87\n",
-      "-86\n",
-      "-85\n",
-      "-84\n",
-      "-83\n",
-      "-82\n",
-      "-81\n",
-      "-80\n",
-      "-79\n",
-      "-78\n",
-      "-77\n",
-      "-76\n",
-      "-75\n",
-      "-74\n",
-      "-73\n",
-      "-72\n",
-      "-71\n",
-      "-70\n",
-      "-69\n",
-      "-68\n",
-      "-67\n",
-      "-66\n",
-      "-65\n",
-      "-64\n",
-      "-63\n",
-      "-62\n",
-      "-61\n",
-      "-60\n",
-      "-59\n",
-      "-58\n",
-      "-57\n",
-      "-56\n",
-      "-55\n",
-      "-54\n",
-      "-53\n",
-      "-52\n",
-      "-51\n",
-      "-50\n",
-      "-49\n",
-      "-48\n",
-      "-47\n",
-      "-46\n",
-      "-45\n",
-      "-44\n",
-      "-43\n",
-      "-42\n"
-     ]
-    },
-    {
-     "ename": "KeyboardInterrupt",
-     "evalue": "",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
-      "\u001b[0;32m<ipython-input-19-221451d1d92a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mecm_trial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mU\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0marange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m100\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m10000\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
-      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mecm_trial\u001b[0;34m(N, arange, krange)\u001b[0m\n\u001b[1;32m     96\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     97\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 98\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     99\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    100\u001b[0m                 \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'try the following: a='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m' and k='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     77\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     78\u001b[0m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 79\u001b[0;31m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     80\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     81\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mtwo_p\u001b[0;34m(x, y, a, b, N)\u001b[0m\n\u001b[1;32m     52\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     53\u001b[0m    \u001b[0;31m# print(Lu)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 54\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     55\u001b[0m   \u001b[0;31m#  print(Lb)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     56\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     43\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     44\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 45\u001b[0;31m     \u001b[0md\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0meuclid\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     46\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0md\u001b[0m\u001b[0;34m==\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
-     ]
-    }
-   ],
-   "source": [
-    "ecm_trial(U,arange=100,krange=10000)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 23,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "1"
-      ]
-     },
-     "execution_count": 23,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "mexp(11,V-1,V)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  }
- ],
- "metadata": {
-  "kernelspec": {
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-    "version": 3
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From e9f7918ce0173306c9c4e33b4abb02c4043ca721 Mon Sep 17 00:00:00 2001
From: Jeremy Teitelbaum <jeremy.teitelbaum@uconn.edu>
Date: Thu, 5 Jul 2018 15:28:31 -0400
Subject: [PATCH 2/2] Delete Untitled.ipynb

---
 ctnt2018/Untitled.ipynb | 1402 ---------------------------------------
 1 file changed, 1402 deletions(-)
 delete mode 100644 ctnt2018/Untitled.ipynb

diff --git a/ctnt2018/Untitled.ipynb b/ctnt2018/Untitled.ipynb
deleted file mode 100644
index 564eeb2..0000000
--- a/ctnt2018/Untitled.ipynb
+++ /dev/null
@@ -1,1402 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 60,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "-20\n",
-      "-19\n",
-      "-18\n",
-      "-17\n",
-      "-16\n",
-      "-15\n",
-      "try the following: a= -15  and k= 8377\n",
-      "-14\n",
-      "-13\n",
-      "-12\n",
-      "try the following: a= -12  and k= 3407\n",
-      "-11\n",
-      "-10\n",
-      "-9\n"
-     ]
-    },
-    {
-     "ename": "KeyboardInterrupt",
-     "evalue": "",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m    115\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    116\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m149185656432189838133\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 117\u001b[0;31m \u001b[0mecm_trial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0marange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m20\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m10000\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    118\u001b[0m \u001b[0;31m#N=2**128+1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    119\u001b[0m \u001b[0;31m#ecm_trial(N,arange=100,krange=10000)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mecm_trial\u001b[0;34m(N, arange, krange)\u001b[0m\n\u001b[1;32m    100\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    101\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 102\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    103\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    104\u001b[0m                 \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'try the following: a='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m' and k='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     81\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     82\u001b[0m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 83\u001b[0;31m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     84\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     85\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mtwo_p\u001b[0;34m(x, y, a, b, N)\u001b[0m\n\u001b[1;32m     56\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     57\u001b[0m    \u001b[0;31m# print(Lu)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 58\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     59\u001b[0m   \u001b[0;31m#  print(Lb)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     60\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     47\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     48\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 49\u001b[0;31m     \u001b[0md\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0meuclid\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     50\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0md\u001b[0m\u001b[0;34m==\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     51\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36meuclid\u001b[0;34m(u, v)\u001b[0m\n\u001b[1;32m     37\u001b[0m         \u001b[0mq\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mx0\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     38\u001b[0m         \u001b[0mx2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mx0\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 39\u001b[0;31m         \u001b[0ma2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0ma0\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0ma1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     40\u001b[0m         \u001b[0mb2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mb0\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mb1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     41\u001b[0m         \u001b[0mx0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb0\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
-     ]
-    }
-   ],
-   "source": [
-    "# %load ECM.py\n",
-    "from math import factorial, gcd, log\n",
-    "import numpy as np\n",
-    "P=np.zeros(10000)\n",
-    "P[0]=1\n",
-    "P[1]=1\n",
-    "for i in range(2,10000):\n",
-    "    if P[i]==0:\n",
-    "        j=2\n",
-    "        while i*j<100:\n",
-    "            P[i*j]=1\n",
-    "            j=j+1\n",
-    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  \n",
-    "B1000=1\n",
-    "B10000=1\n",
-    "for x in Primes10000:\n",
-    "    if x<1000:\n",
-    "        B1000=B1000*(x**(int(log(1000)/log(x))))\n",
-    "    B10000=B10000*(x**(int(log(1000)/log(x))))\n",
-    "\n",
-    "def mexp(a,x,N):\n",
-    "    m,s=1,a\n",
-    "    while x>0:\n",
-    "        if x % 2 ==1:\n",
-    "            m=((m*s) % N)\n",
-    "        s=((s*s) % N)\n",
-    "        x=x//2\n",
-    "    return m\n",
-    "\n",
-    "def euclid(u,v):\n",
-    "    if v==0:\n",
-    "        raise ArithmeticError('Division by Zero')\n",
-    "    x0,x1=u,v\n",
-    "    a0,a1=1,0\n",
-    "    b0,b1=0,1\n",
-    "    while x1!=0:\n",
-    "        q=x0//x1\n",
-    "        x2=x0-q*x1\n",
-    "        a2=a0-q*a1\n",
-    "        b2=b0-q*b1\n",
-    "        x0,a0,b0=x1,a1,b1\n",
-    "        x1,a1,b1=x2,a2,b2\n",
-    "    if x0<0:\n",
-    "        return -x0,-a0,-b0\n",
-    "    else:\n",
-    "        return x0,a0,b0\n",
-    "\n",
-    "def mod_inv(u,N):\n",
-    "    d,a,b=euclid(u,N)\n",
-    "    if d==1:\n",
-    "        return a\n",
-    "    else:\n",
-    "        raise ArithmeticError('Common factor is '+str(d))\n",
-    "\n",
-    "def two_p(x,y,a,b,N):\n",
-    "    Lu=(3*x**2+a) % N\n",
-    "   # print(Lu)\n",
-    "    Lb=mod_inv(2*y,N)\n",
-    "  #  print(Lb)\n",
-    "    L=Lu*Lb % N\n",
-    "    x_two=(L*L-2*x) % N\n",
-    "    y_two=(L*(x-x_two)-y) %N\n",
-    "    return x_two,y_two\n",
-    "\n",
-    "def sum_p(x1,y1,x2,y2,a,b,N):\n",
-    "    Lu=(y2-y1) % N\n",
-    "    Lb=mod_inv(x2-x1,N)\n",
-    "    L=(Lu*Lb) % N\n",
-    "    x_sum=(L*L-x1-x2) %N\n",
-    "    y_sum=(L*(x1-x_sum)-y1) %N\n",
-    "    return x_sum,y_sum\n",
-    "    \n",
-    "def exp_p(x,y,a,b,m,N):\n",
-    "    sx,sy=x,y\n",
-    "    first=True\n",
-    "    while m>0:\n",
-    "        if m%2==1:\n",
-    "            if first:\n",
-    "                xm,ym=sx,sy\n",
-    "                first=False\n",
-    "            else:\n",
-    "                xm,ym=sum_p(xm,ym,sx,sy,a,b,N)\n",
-    "        sx,sy=two_p(sx,sy,a,b,N)\n",
-    "        m=m//2\n",
-    "    return xm,ym\n",
-    "\n",
-    "#def mexp(a,x,N):\n",
-    " #   m,s=1,a\n",
-    " #   while x>0:\n",
-    " #       if x % 2 ==1:\n",
-    " #           m=((m*s) % N)\n",
-    " #       s=((s*s) % N)\n",
-    " #       x=x//2\n",
-    " #   return m\n",
-    "\n",
-    "def ecm_trial(N,arange=50,krange=30):\n",
-    "    for a in range(-arange,arange):\n",
-    "        xm,ym=0,1\n",
-    "        print(a) \n",
-    "        for k in range(2,krange):\n",
-    "            try:\n",
-    "                xm,ym=exp_p(xm,ym,a,1,k,N)\n",
-    "            except ArithmeticError:\n",
-    "                print('try the following: a=',a,' and k=',k)\n",
-    "                break\n",
-    "\n",
-    "def ecm_trial_2(N,arange=50,B=B1000):\n",
-    "    for a in range(-arange,arange):\n",
-    "        xm,ym=0,1\n",
-    "        try:\n",
-    "            xm,ym=exp_p(xm,ym,a,1,B,N)\n",
-    "        except ArithmeticError:\n",
-    "            print('try the following: a=',a)\n",
-    "            break\n",
-    "                    \n",
-    "N=149185656432189838133\n",
-    "ecm_trial(N,arange=20,krange=10000)\n",
-    "#N=2**128+1\n",
-    "#ecm_trial(N,arange=100,krange=10000)\n",
-    "#xm,ym=exp_p(0,1,-91,1,factorial(7883),N)\n",
-    "                                                                                                                                    \n",
-    "                                            \n",
-    "\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 61,
-   "metadata": {},
-   "outputs": [
-    {
-     "ename": "ArithmeticError",
-     "evalue": "Common factor is 16029560051",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mArithmeticError\u001b[0m                           Traceback (most recent call last)",
-      "\u001b[0;32m<ipython-input-61-1561928d1db5>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m15\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mfactorial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m8377\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     80\u001b[0m                 \u001b[0mfirst\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     81\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 82\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     83\u001b[0m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     84\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36msum_p\u001b[0;34m(x1, y1, x2, y2, a, b, N)\u001b[0m\n\u001b[1;32m     65\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     66\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 67\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     68\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     69\u001b[0m     \u001b[0mx_sum\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     51\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     52\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 53\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Common factor is '\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     54\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     55\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;31mArithmeticError\u001b[0m: Common factor is 16029560051"
-     ]
-    }
-   ],
-   "source": [
-    "exp_p(0,1,-15,1,factorial(8377),N)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 57,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "try the following: a= -91\n"
-     ]
-    }
-   ],
-   "source": [
-    "P=np.zeros(10000)\n",
-    "P[0]=1\n",
-    "P[1]=1\n",
-    "for i in range(2,10000):\n",
-    "    if P[i]==0:\n",
-    "        j=2\n",
-    "        while i*j<100:\n",
-    "            P[i*j]=1\n",
-    "            j=j+1\n",
-    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  \n",
-    "B1000=1\n",
-    "B10000=1\n",
-    "for x in Primes10000:\n",
-    "    if x<1000:\n",
-    "        B1000=B1000*(x**(int(log(1000)/log(x))))\n",
-    "    B10000=B10000*(x**(int(log(10000)/log(x))))\n",
-    "ecm_trial_2(N,arange=100,B=B10000)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 31,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "B=1\n",
-    "for x in Primes10000:\n",
-    "    B=B*(x**(int(log(1000000)/log(x))))\n",
-    "    "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 51,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "(272092429413788558381560766739333595442,\n",
-       " 304592951138784758364937297795729494958)"
-      ]
-     },
-     "execution_count": 51,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "exp_p(0,1,-91,1,B10000,N)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 32,
-   "metadata": {},
-   "outputs": [
-    {
-     "ename": "ArithmeticError",
-     "evalue": "Common factor is 59649589127497217",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mArithmeticError\u001b[0m                           Traceback (most recent call last)",
-      "\u001b[0;32m<ipython-input-32-ea587d3a9f9d>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m91\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mB\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
-      "\u001b[0;32m<ipython-input-26-e41ba6d1881a>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     76\u001b[0m                 \u001b[0mfirst\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     77\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 78\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     79\u001b[0m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     80\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-26-e41ba6d1881a>\u001b[0m in \u001b[0;36msum_p\u001b[0;34m(x1, y1, x2, y2, a, b, N)\u001b[0m\n\u001b[1;32m     61\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     62\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 63\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     64\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     65\u001b[0m     \u001b[0mx_sum\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-26-e41ba6d1881a>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     48\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 49\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Common factor is '\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     50\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     51\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;31mArithmeticError\u001b[0m: Common factor is 59649589127497217"
-     ]
-    }
-   ],
-   "source": [
-    "exp_p(0,1,-91,1,B,N)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Primes1000"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 29,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "from math import factorial, gcd, log\n",
-    "import numpy as np\n",
-    "P=np.zeros(10000)\n",
-    "P[0]=1\n",
-    "P[1]=1\n",
-    "for i in range(2,10000):\n",
-    "    if P[i]==0:\n",
-    "        j=2\n",
-    "        while i*j<10000:\n",
-    "            P[i*j]=1\n",
-    "            j=j+1\n",
-    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 30,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
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-       " 6257,\n",
-       " 6263,\n",
-       " 6269,\n",
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-       " ...]"
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