project_euler.problem_136.sol1
==============================

.. py:module:: project_euler.problem_136.sol1

.. autoapi-nested-parse::

   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
---------

.. autoapisummary::

   project_euler.problem_136.sol1.solution


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

.. py:function:: 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