project_euler.problem_136.sol1

Project Euler Problem 136: https://projecteuler.net/problem=136

Singleton Difference

The positive integers, x, y, and z, are consecutive terms of an arithmetic progression. Given that n is a positive integer, the equation, x^2 - y^2 - z^2 = n, has exactly one solution when n = 20:

13^2 - 10^2 - 7^2 = 20.

In fact there are twenty-five values of n below one hundred for which the equation has a unique solution.

How many values of n less than fifty million have exactly one solution?

By change of variables

x = y + delta z = y - delta

The expression can be rewritten:

x^2 - y^2 - z^2 = y * (4 * delta - y) = n

The algorithm loops over delta and y, which is restricted in upper and lower limits, to count how many solutions each n has. In the end it is counted how many n’s have one solution.

Functions

solution(→ int)

Define n count list and loop over delta, y to get the counts, then check

Module Contents

project_euler.problem_136.sol1.solution(n_limit: int = 50 * 10**6) → int

Define n count list and loop over delta, y to get the counts, then check which n has count == 1.

>>> solution(3)
0
>>> solution(10)
3
>>> solution(100)
25
>>> solution(110)
27