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 \[N_2(K) ≤ 2K(K+1)+1.\] Via an argument of Granville's we additionally deduce the existence of infinitely many non-trivial prime valued K-multimagic squares of order $2K(K+1)+1$.