  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.
