eulerbooks/notebooks/problem0039.ipynb

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{
"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": {
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"display_name": "SageMath 9.5",
"language": "sage",
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