K-multimagic squares and magic squares of k-th powers via the circle method

Abstract: Here we investigate K-multimagic squares of order N. These are N x 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 N₂(K) for which there exist nontrivial K-multimagic squares of order N₂(K). 

Previous results on multimagic squares show that N₂(K) ≤  (4K-2)^K for large K. We use the Hardy-Littlewood circle method to improve this to 

N₂ (K) ≤ 2K(K+1)+1

The intricate structure of the coefficient matrix poses significant technical challenges for the circle method. We overcome these obstacles by generalizing the class of Diophantine systems amenable to the circle method and demonstrating that the multimagic square system belongs to this class for all N ≥ 4. We additionally extend our results to magic squares consisting of K^th powers.

 

Host: Alan Chang