Given the following square, where $a,b,c,d \in \mathbb{N}$:
$$\left[\begin{matrix} a&b\\ c&d \end{matrix}\right]$$
How do you find the set of $\mathbb{N}$ that satisfies the following condition: $$a^n+b^n=c^n+d^n=a^n+c^n=b^n+d^n=c^n+b^n=a^n+d^n$$ where $n\geq2,\in\mathbb{N}$.
Starting small, does a solution exist for $n=2$? Aside from trial and error, what method should I use to even begin?
I am aware that you cannot construct a magic square with only 4 terms, but what if the terms were squared? cubed? Does such a limitation extend to all powers?
I think it just has a trivial solution: $a=b=c=d$.