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* [pre-commit.ci] pre-commit autoupdate

updates:
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* updating DIRECTORY.md

* Fix typos in test_min_spanning_tree_prim.py

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Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
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pre-commit-ci[bot]
2023-10-07 21:32:28 +02:00
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30
project_euler/problem_135/sol1.py
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@ -1,28 +1,22 @@
"""
Project Euler Problem 135: https://projecteuler.net/problem=135
Given the positive integers, x, y, and z,
are consecutive terms of an arithmetic progression,
the least value of the positive integer, n,
for which the equation,
Given the positive integers, x, y, and z, are consecutive terms of an arithmetic
progression, the least value of the positive integer, n, for which the equation,
x2 − y2 − z2 = n, has exactly two solutions is n = 27:
342 − 272 − 202 = 122 − 92 − 62 = 27
It turns out that n = 1155 is the least value
which has exactly ten solutions.
It turns out that n = 1155 is the least value which has exactly ten solutions.
How many values of n less than one million
have exactly ten distinct solutions?
How many values of n less than one million have exactly ten distinct solutions?
Taking x,y,z of the form a+d,a,a-d respectively,
the given equation reduces to a*(4d-a)=n.
Calculating no of solutions for every n till 1 million by fixing a
,and n must be multiple of a.
Total no of steps=n*(1/1+1/2+1/3+1/4..+1/n)
,so roughly O(nlogn) time complexity.
Taking x, y, z of the form a + d, a, a - d respectively, the given equation reduces to
a * (4d - a) = n.
Calculating no of solutions for every n till 1 million by fixing a, and n must be a
multiple of a. Total no of steps = n * (1/1 + 1/2 + 1/3 + 1/4 + ... + 1/n), so roughly
O(nlogn) time complexity.
"""
@ -42,15 +36,15 @@ def solution(limit: int = 1000000) -> int:
for first_term in range(1, limit):
for n in range(first_term, limit, first_term):
common_difference = first_term + n / first_term
if common_difference % 4: # d must be divisble by 4
if common_difference % 4: # d must be divisible by 4
continue
else:
common_difference /= 4
if (
first_term > common_difference
and first_term < 4 * common_difference
): # since x,y,z are positive integers
frequency[n] += 1 # so z>0 and a>d ,also 4d<a
): # since x, y, z are positive integers
frequency[n] += 1 # so z > 0, a > d and 4d < a
count = sum(1 for x in frequency[1:limit] if x == 10)