add problem 58

notebooks/problem0058.ipynb

@@ -0,0 +1,91 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "7edea8e5",
+   "metadata": {},
+   "source": [
+    "# [Spiral Primes](https://projecteuler.net/problem=58)\n",
+    "\n",
+    "It's the return of the Ulam spiral from [problem 28](https://projecteuler.net/problem=28) (this time we're going counter-clockwise, but that doesn't actually affect much).\n",
+    "\n",
+    "We can handle this problem with a couple of easy-to-derive formulas. First, for a spiral with side length $n$ (note that $n$ must be odd), the number of diagonal entries is $2n-1$. Furthermore, the outermost diagonal entries will be $n^2$, $n^2 - (n-1)$, $n^2 - 2(n-1)$, and $n^2 - 3(n-1)$.\n",
+    "\n",
+    "With these facts, we can just iterate over odd values of $n$ and calculate the four outermost diagonal entries. We'll keep a running total $p$ of how many primes we see and stop when $\\frac{p}{2n-1} < 0.1$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "4480be30",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def diagonal(n, k): return n^2 - k*(n-1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "025d7d4b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "26241"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from itertools import count\n",
+    "\n",
+    "p = 0\n",
+    "for n in count(3, 2):\n",
+    "    for k in range(0, 4):\n",
+    "        if is_prime(diagonal(n, k)):\n",
+    "            p += 1\n",
+    "            \n",
+    "    if p / (2*n - 1) < 0.1:\n",
+    "        break\n",
+    "\n",
+    "n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f181c6ae",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Numbers on diagonals: [A200975](https://oeis.org/A200975)\n",
+    "* Primes at right-angle turns on the Ulam spiral: [A172979](https://oeis.org/A172979)"
+   ]
+  }
+ ],
+ "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
+}