mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-10-06 13:49:30 +00:00
62 lines
1.7 KiB
Python
62 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 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() = }")
|