{
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   "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",
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  "tags": [
   "prime",
   "goldbach",
   "composite",
   "square",
   "fermat"
  ]
 },
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}