From 07b599000e709d066a41ab0d3b87ec9fd5125503 Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Wed, 18 Jun 2025 00:06:53 -0400
Subject: [PATCH] add problem 75

---
 notebooks/problem0075.ipynb | 130 ++++++++++++++++++++++++++++++++++++
 1 file changed, 130 insertions(+)
 create mode 100644 notebooks/problem0075.ipynb

diff --git a/notebooks/problem0075.ipynb b/notebooks/problem0075.ipynb
new file mode 100644
index 0000000..02b9589
--- /dev/null
+++ b/notebooks/problem0075.ipynb
@@ -0,0 +1,130 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "eb8a199b",
+   "metadata": {},
+   "source": [
+    "# [Singular Integer Right Triangles](https://projecteuler.net/problem=75)\n",
+    "\n",
+    "This problem is eerily similar to [problem 39](https://projecteuler.net/problem=39). We'll reuse our generator from that problem:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "9d02cd79",
+   "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": "3d3e6bf8",
+   "metadata": {},
+   "source": [
+    "And we'll reuse the main loop from that problem, too!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "09e14708",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from collections import Counter\n",
+    "\n",
+    "limit = 1500000\n",
+    "lengths = Counter()\n",
+    "\n",
+    "for (a, b, c) in primitive_pythagorean_triplets(limit):\n",
+    "    for k in count(1):\n",
+    "        L = k * (a + b + c)\n",
+    "        if L > limit:\n",
+    "            break\n",
+    "        \n",
+    "        lengths[L] += 1"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c12a3e3c",
+   "metadata": {},
+   "source": [
+    "The only differences in this problem are our maximum perimeter, and that we're looking for perimeters that can only be made from one Pythagorean triplet."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "0e7ce4b0",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "161667"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "len({k for (k, v) in lengths.items() if v == 1})"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84d83e3b",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Perimeters with one Pythagorean triple: [A098714](https://oeis.org/A098714)"
+   ]
+  }
+ ],
+ "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
+}