A Pythagorean triangle in which the hypotenuse and the sum of the arms are squares

TL;DR

This paper investigates a specific Diophantine equation related to Pythagorean triangles with square hypotenuse and sum of legs, proving only two solutions exist and developing an algorithm to find such solutions.

Contribution

The paper proves the finiteness of solutions to a classic problem and introduces an algorithm to find primitive solutions, expanding understanding of Pythagorean triangles with special properties.

Findings

Only two solutions to the equation exist: (0,1) and (119,13).

The relationship between primitive Pythagorean triples and Pell's equation is elucidated.

An algorithm for finding primitive solutions is presented.

Abstract

In this paper, show that the Diophantine equation $ x^2+(x+1)^2=w^4 $ has only two solutions $ (0,1) $ and $ (119,13)$ in non-negative integers $ x $ and $ w $. This equation concerned a classic problem posed by Pierre de Fermat, wonders about finding a Pythagorean triangle in which the hypotenuse and the sum of the arms are square. We review the method of finding the smallest solution presented by Fermat, and the relationship between the primitive Pythagorean triples and the Pell's equation, Finally, we present an algorithm for finding primitive solutions, which actually enabled us to find other solutions.