forked from YovM/Algorithms
56 lines
1.8 KiB
Python
56 lines
1.8 KiB
Python
"""
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Project Euler Problem 135: https://projecteuler.net/problem=135
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Given the positive integers, x, y, and z, are consecutive terms of an arithmetic
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progression, the least value of the positive integer, n, for which the equation,
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x2 − y2 − z2 = n, has exactly two solutions is n = 27:
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342 − 272 − 202 = 122 − 92 − 62 = 27
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It turns out that n = 1155 is the least value which has exactly ten solutions.
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How many values of n less than one million have exactly ten distinct solutions?
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Taking x, y, z of the form a + d, a, a - d respectively, the given equation reduces to
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a * (4d - a) = n.
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Calculating no of solutions for every n till 1 million by fixing a, and n must be a
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multiple of a. Total no of steps = n * (1/1 + 1/2 + 1/3 + 1/4 + ... + 1/n), so roughly
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O(nlogn) time complexity.
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"""
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def solution(limit: int = 1000000) -> int:
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"""
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returns the values of n less than or equal to the limit
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have exactly ten distinct solutions.
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>>> solution(100)
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0
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>>> solution(10000)
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45
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>>> solution(50050)
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292
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"""
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limit = limit + 1
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frequency = [0] * limit
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for first_term in range(1, limit):
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for n in range(first_term, limit, first_term):
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common_difference = first_term + n / first_term
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if common_difference % 4: # d must be divisible by 4
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continue
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else:
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common_difference /= 4
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if (
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first_term > common_difference
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and first_term < 4 * common_difference
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): # since x, y, z are positive integers
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frequency[n] += 1 # so z > 0, a > d and 4d < a
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count = sum(1 for x in frequency[1:limit] if x == 10)
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return count
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if __name__ == "__main__":
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print(f"{solution() = }")
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