Python/project_euler/problem_135/sol1.py

"""
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 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 divisble 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

    count = sum(1 for x in frequency[1:limit] if x == 10)

    return count


if __name__ == "__main__":
    print(f"{solution() = }")