A circle method approach to K-multimagic squares

In this paper we investigate K-multimagic squares of order N, these are N N magic squares which remain magic after raising each element to the kth power for all 2 k K. Given K 2, we consider the problem of establishing the smallest integer N₂ (K) for which there exists non-trivial K-multimagic squares of order N₂ (K). Previous results on multimagic squares show that N₂ (K) (4K-2) K for large K. Here we utilize the Hardy-Littlewood circle method and establish the bound ₂ (K) 2K (K+1) +1. \ Via an argument of Granville's we additionally deduce the existence of infinitely many non-trivial prime valued K-multimagic squares of order 2K (K+1) +1.