Python/project_euler/problem_135/sol1.py
Maxim Smolskiy 4700297b3e
Enable ruff RUF002 rule ()
* Enable ruff RUF002 rule

* Fix

---------

Co-authored-by: Christian Clauss <cclauss@me.com>
2024-04-22 21:51:47 +02:00

56 lines
1.7 KiB
Python

"""
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,
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.
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 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.
"""
def solution(limit: int = 1000000) -> int:
"""
returns the values of n less than or equal to the limit
have exactly ten distinct solutions.
>>> solution(100)
0
>>> solution(10000)
45
>>> solution(50050)
292
"""
limit = limit + 1
frequency = [0] * limit
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 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, a > d and 4d < a
count = sum(1 for x in frequency[1:limit] if x == 10)
return count
if __name__ == "__main__":
print(f"{solution() = }")