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

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.