{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "42fd0b2c",
   "metadata": {},
   "source": [
    "# [Goldbach's Other Conjecture](https://projecteuler.net/problem=46)\n",
    "\n",
    "We can write a function that takes odd composites $m$ and checks successively larger values of $n$ to see if $m - 2n^2$ is prime. If we find such an $n$, $m$ satisfies the conjecture, but if $n$ gets too large, $m - 2n^2$ will become non-positive, and therefore can't be prime, so $m$ fails to satisfy the conjecture."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "eba09d52",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5777"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from itertools import count\n",
    "\n",
    "def satisfies_conjecture(m):\n",
    "    for n in count(1):\n",
    "        p = m - 2 * n^2\n",
    "        if p <= 0:\n",
    "            return False\n",
    "        \n",
    "        if is_prime(p):\n",
    "            return True\n",
    "\n",
    "\n",
    "for k in count(2):\n",
    "    m = 2 * k - 1\n",
    "    if is_prime(m):\n",
    "        continue\n",
    "        \n",
    "    if not satisfies_conjecture(m):\n",
    "        break\n",
    "\n",
    "m"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd35bb73",
   "metadata": {},
   "source": [
    "Interestingly, the only other number known not to satisfy this conjecture is 5993."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3412975d",
   "metadata": {},
   "source": [
    "## Relevant sequences\n",
    "* Stern numbers (includes all odd numbers, not just composites): [A060003](https://oeis.org/A060003)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.5",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}