add problem 46

notebooks/problem0046.ipynb

@@ -0,0 +1,93 @@
+{
+ "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)"
+   ]
+  }
+ ],
+ "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
+}