A circle method approach to K‐multimagic squares

In this paper, we investigate ‐multimagic squares of order . These are magic squares that remain magic after raising each element to the th power for all . Given , we consider the problem of establishing the smallest integer for which there exist nontrivial ‐multimagic squares of order . Previous results on multimagic squares show that for large . We use the Hardy–Littlewood circle method to improve this to 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 .