solving a Diophantine system of two equations Find all triples $(a,b,c) \in \mathbb{N}$ satisfing the following equations:-
\begin{align}
a^2 + b^2 & = c^3 \\
(a + b)^2 & = c^4
\end{align}
Thank you for your help.

Hint: Since $(a^2+b^2)c=c^4=(a+b)^2$ we have that $a^2+b^2$ divides $(a+b)^2$. This implies $a=b$ for positive integers. Then we have $2a^2=c^3$, so that $(a,b,c)=(2,2,2)$.
