project_euler.problem_138.sol1

Project Euler Problem 138: https://projecteuler.net/problem=138

Special Isosceles Triangles

With change of variables

c = b/2

and requiring that

h = 2c +- 1

the triangle relation

c^2 + h^2 = L^2

can be expressed as

5 c^2 +- 4c + 1 = L^2

or with some rearrangement:

(5c +- 2)^2 = 5L^2 - 1

This to be solved for positive integer c and L, requires that

5L^2 - 1 = m^2

The above equation is negative Pell’s equation with n = 5 and can be solved recursively as outlined in the wikipedia article. Note, we neglect first solution (m = 2, L = 1), as this leads to b and h being non-integers.

Reference: https://en.wikipedia.org/wiki/Pell%27s_equation#The_negative_Pell’s_equation

Functions

solution(→ int)

The recursive solution of negative Pell's equation with k + 1 values of L

Module Contents

project_euler.problem_138.sol1.solution(k: int = 12) → int

The recursive solution of negative Pell’s equation with k + 1 values of L summed and the first solution being skipped.

>>> solution(2)
322
>>> solution(5)
1866293