mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-12-21 02:30:15 +00:00
4700297b3e
* Enable ruff RUF002 rule * Fix --------- Co-authored-by: Christian Clauss <cclauss@me.com>
56 lines
1.7 KiB
Python
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() = }")
|