A circle method approach to K-multimagic squares

In this paper we investigate $K$-multimagic squares of order $N$, these are $N \times N$ magic squares which remain magic after raising each element to the $k$ th power for all $2 \leqslant$ $k \leqslant K$. Given $K \geqslant 2$, we consider the problem of establishing the smallest integer $N_2(K)$ for which there exists nontrivial $K$-multimagic squares of order $N_2(K)$. Previous results on multimagic squares show that $N_2(K) \leqslant(4 K-2)^K$ for large $K$. Here we utilize the Hardy-Littlewood circle method and establish the bound $$N_2(K) \leqslant 2 K(K+1)+1$$ Via an argument of Granville's we additionally deduce the existence of infinitely many nontrivial prime valued $K$-multimagic squares of order $2 K(K+1)+1$.