A circle method approach to K-multimagic squares

2024/06/12 by Daniel Flores, Flores, Daniel
Computer Science · Decision Sciences · #05B15 #05B20 #11D45 #11D72 #11E76 #11L07 #11P55 #Combinatorics (math.CO) #FOS: Mathematics #Fuzzy and Soft Set Theory #Graph Labeling and Dimension Problems #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.08161

openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 k th power for all 2 \leqslant k \leqslant K. Given K \geqslant 2, we consider the problem of establishing the smallest integer N₂(K) for which there exists nontrivial K-multimagic squares of order N₂(K). Previous results on multimagic squares show that N₂(K) \leqslant(4 K-2)^K for large K. Here we utilize the Hardy-Littlewood circle method and establish the bound N₂(K) \leqslant 2 K(K+1)+1 Via an argument of Granville's we additionally deduce the existence of infinitely many nontrivial prime valued K-multimagic squares of order 2 K(K+1)+1.

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