Solved 46.

ipython/EulerProblem045.ipynb

@@ -31,7 +31,9 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "def T(n):\n",
@@ -53,7 +55,9 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "n_max = 100000\n",
@@ -119,7 +123,7 @@
   },
   "tags": [
    "triangular",
-   "pentagona",
+   "pentagonal",
    "hexagonal"
   ]
  },

ipython/EulerProblem046.ipynb

@@ -0,0 +1,156 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Goldbach's other conjecture (Euler Problem 46)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "collapsed": true
+   },
+   "source": [
+    "[https://projecteuler.net/problem=46](https://projecteuler.net/problem=46)\n",
+    "\n",
+    "It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.\n",
+    "\n",
+    "$9 = 7 + 2×1^2$\n",
+    "\n",
+    "$15 = 7 + 2×2^2$\n",
+    "\n",
+    "$21 = 3 + 2×3^2$\n",
+    "\n",
+    "$25 = 7 + 2×3^2$\n",
+    "\n",
+    "$27 = 19 + 2×2^2$\n",
+    "\n",
+    "$33 = 31 + 2×1^2$\n",
+    "\n",
+    "It turns out that the conjecture was false.\n",
+    "\n",
+    "What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Okay, we reuse Fermat's test and brute force. Easy."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "def expmod(base, exp, m):\n",
+    "    if exp == 0:\n",
+    "        return 1\n",
+    "    if (exp % 2 == 0):\n",
+    "        return (expmod(base, exp // 2, m) ** 2 % m)\n",
+    "    return (base * expmod(base, exp - 1, m) % m)\n",
+    "\n",
+    "def fermat_test(n):\n",
+    "    a = n - 3\n",
+    "    return expmod(a, n, n) == a"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "def twice_square(n):\n",
+    "    return 2 * n * n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n_max = 10000\n",
+    "twice_squares = [twice_square(n) for n in range(1, n_max + 1)]\n",
+    "\n",
+    "def test_conjecture(n):\n",
+    "    for ts in twice_squares:\n",
+    "        if ts > n:\n",
+    "            return False\n",
+    "        if fermat_test(n - ts):\n",
+    "            return True\n",
+    "        \n",
+    "assert(test_conjecture(33))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "5777\n"
+     ]
+    }
+   ],
+   "source": [
+    "for n in range(3, n_max + 1, 2):\n",
+    "    if not fermat_test(n) and test_conjecture(n) == False:\n",
+    "        s = n\n",
+    "\n",
+    "print(s)\n",
+    "assert(s == 5777)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "completion_date": "Sat, 22 Dec 2018, 23:39",
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python3.6",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.6.5"
+  },
+  "tags": [
+   "prime",
+   "goldbach",
+   "composite",
+   "square",
+   "fermat"
+  ]
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}