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62 lines
1.7 KiB
Python
62 lines
1.7 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,
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are consecutive terms of an arithmetic progression,
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the least value of the positive integer, n,
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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
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which has exactly ten solutions.
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How many values of n less than one million
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have exactly ten distinct solutions?
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Taking x,y,z of the form a+d,a,a-d respectively,
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the given equation reduces to a*(4d-a)=n.
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Calculating no of solutions for every n till 1 million by fixing a
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,and n must be multiple of a.
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Total no of steps=n*(1/1+1/2+1/3+1/4..+1/n)
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,so roughly 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 divisble 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 and a>d ,also 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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