135 lines
3.2 KiB
Plaintext
135 lines
3.2 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "eb8a199b",
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"metadata": {},
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"source": [
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"# [Singular Integer Right Triangles](https://projecteuler.net/problem=75)\n",
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"\n",
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"This problem is eerily similar to [problem 39](https://projecteuler.net/problem=39). We'll reuse our generator from that problem:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "9d02cd79",
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"metadata": {},
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"outputs": [],
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"source": [
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"from itertools import count\n",
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"\n",
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"def primitive_pythagorean_triplets(max_perim):\n",
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" for m in count(2):\n",
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" if 2*m^2 + 2*m > max_perim:\n",
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" break\n",
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"\n",
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" for n in range(1, m):\n",
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" if not ((m % 2) != (n % 2)) or gcd(m, n) != 1:\n",
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" continue\n",
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" \n",
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" a = m^2 - n^2\n",
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" b = 2*m*n\n",
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" c = m^2 + n^2\n",
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" \n",
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" if a + b + c > max_perim:\n",
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" break\n",
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" \n",
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" yield (a, b, c)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3d3e6bf8",
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"metadata": {},
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"source": [
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"And we'll reuse the main loop from that problem, too!"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "09e14708",
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"metadata": {},
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"outputs": [],
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"source": [
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"from collections import Counter\n",
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"\n",
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"limit = 1500000\n",
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"lengths = Counter()\n",
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"\n",
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"for (a, b, c) in primitive_pythagorean_triplets(limit):\n",
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" for k in count(1):\n",
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" L = k * (a + b + c)\n",
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" if L > limit:\n",
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" break\n",
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" \n",
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" lengths[L] += 1"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c12a3e3c",
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"metadata": {},
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"source": [
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"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."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "0e7ce4b0",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"161667"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"len({k for (k, v) in lengths.items() if v == 1})"
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]
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},
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{
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"cell_type": "markdown",
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"id": "84d83e3b",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Perimeters with one Pythagorean triple: [A098714](https://oeis.org/A098714)\n",
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"\n",
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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"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)."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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