project_euler.problem_138.sol1
==============================

.. py:module:: project_euler.problem_138.sol1

.. autoapi-nested-parse::

   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
---------

.. autoapisummary::

   project_euler.problem_138.sol1.solution


Module Contents
---------------

.. py:function:: 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