add problem 39

notebooks/problem0039.ipynb

@@ -0,0 +1,115 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "78c9390b",
+   "metadata": {},
+   "source": [
+    "# [Integer Right Triangles](https://projecteuler.net/problem=39)\n",
+    "\n",
+    "If a right triangle has integer side lengths, the side lengths are a [Pythagorean triple](https://en.wikipedia.org/wiki/Pythagorean_triple). In [problem 9](https://projecteuler.net/problem=9), we wrote a generator for primitive Pythagorean triples based off of Euclid's formula. We can modify that generator to cut off after the triplets have passed a maximum perimeter. Note that the perimeter of a triangle with side lengths generated by Euclid's formula will have perimeter $2m^2 + 2mn$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "b03eb872",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from itertools import count\n",
+    "\n",
+    "def primitive_pythagorean_triplets(max_perim):\n",
+    "    for m in count(2):\n",
+    "        if 2*m^2 + 2*m > max_perim:\n",
+    "            break\n",
+    "\n",
+    "        for n in range(1, m):\n",
+    "            if not ((m % 2) ^^ (n % 2)) or gcd(m, n) != 1:\n",
+    "                continue\n",
+    "            \n",
+    "            a = m^2 - n^2\n",
+    "            b = 2*m*n\n",
+    "            c = m^2 + n^2\n",
+    "            \n",
+    "            if a + b + c > max_perim:\n",
+    "                break\n",
+    "            \n",
+    "            yield (a, b, c)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "431ef21c",
+   "metadata": {},
+   "source": [
+    "Now we can just iterate through our new generator and count the perimeters. We also count multiples of the perimeters to include non-primitive triplets."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "62aa955f",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "840"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from collections import Counter\n",
+    "\n",
+    "max_perim = 1000\n",
+    "perimeters = Counter()\n",
+    "for (a, b, c) in primitive_pythagorean_triplets(max_perim):\n",
+    "    for k in count(1):\n",
+    "        perimeter = k * (a + b + c)\n",
+    "        if perimeter > max_perim:\n",
+    "            break\n",
+    "        \n",
+    "        perimeters[perimeter] += 1\n",
+    "\n",
+    "max(perimeters, key=perimeters.get)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6cb1b692",
+   "metadata": {},
+   "source": [
+    "## Related sequences\n",
+    "* Number of integer right triangles with perimeter $n$: [A024155](https://oeis.org/A024155)"
+   ]
+  }
+ ],
+ "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
+}