<s>[INST] 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.
 [/INST] We have that 
$$
c=\frac{(a+b)^2}{a^2+b^2}=\frac{a^2+2ab+b^2}{a^2+b^2}=1+\frac{2ab}{a^2+b^2}, 
$$ 
so $\dfrac{2ab}{a^2+b^2}$  must be an integer. But by Cauchy inequality $2ab \leq a^2+b^2$ and an integer can be only if $2ab=a^2+b^2$. It implies that  that $a=b$ and $c=2$.
From the first equation we have $2 a^2=c^3=8 \implies a=2.$
Se we get the one solution $(2,2,2).$
If $0 \in \mathbb{N}$ then we have trivial  solution $(0, 0,0).$ Also there are two more cases when $2ab \leq a^2+b^2$ is integer for $ a=0 \implies b=1$ and for $b=0 \implies a=1.$ So we get also two solutions $(0,1,1)$ and $(1,0,1).$
</s>