Add solution for the euler project problem 138.

Author: mindaugl

project_euler/problem_138/__init__.py

@@ -0,0 +1,63 @@
+"""
+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
+
+"""
+
+
+def 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
+    """
+
+    m_i = 2
+    l_i = 1
+    ans = 0
+    for _ in range(2, k + 2):
+        m_i, l_i = 4 * m_i + 5 * m_i + 20 * l_i, 4 * l_i + 5 * l_i + 4 * m_i
+        ans += l_i
+
+    return ans
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")