K-multimagic squares and magic squares of kth powers via the circle method.

Abstract: 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 N(K) for which there exists non-trivial K-multimagic squares of order N(K). Previous results on multimagic squares show that N(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. We additionally address the simpler problem of magic squares consisting of kth powers, improving on a recent result by Rome and Yamagishi.