A circle method approach to K-multimagic squares

Abstract

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 $2⩽$ $k⩽K$. Given $K⩾2$, 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 $N_2(K)⩽(4K-2{)}^K$ for large $K$. Here we utilize the Hardy-Littlewood circle method and establish the bound $$N_2(K)⩽2K(K+1)+1$$ Via an argument of Granville's we additionally deduce the existence of infinitely many nontrivial prime valued $K$-multimagic squares of order $2K(K+1)+1$.

Fixed a mistake in the proof of Lemma 4.1