A circle method approach to K-multimagic squares

TL;DR

This work addresses the existence threshold for nontrivial K-multimagic squares and proves an explicit upper bound $N_2(K)\le 2K(K+1)+1$ using the Hardy–Littlewood circle method. By encoding MMS constraints into a diagonal system with a carefully constructed matrix $C^{\text{magic}}_{N}$ and establishing a domination property against a function $F$, the authors obtain the asymptotic count $M_{K,N}(P)\sim cP^{N(N-K(K+1))}$ for $N>2K(K+1)$ and thereby guarantee nontrivial MMS$(K,N)$ existence in this range; Granville's argument further yields infinitely many prime-valued MMS of order $2K(K+1)+1$. The analysis hinges on a rigorous major/minor arc decomposition, absolute convergence and positivity of the singular series and integral under nonsingular local solvability, and a constructive local-solvability argument via DDLS$(N)$. Overall, the circle-method framework produces a quadratic-in-$K$ growth bound for the minimal order and opens pathways to higher-dimensional generalizations and prime-valued analogues. The results substantially improve prior exponential bounds and demonstrate the feasibility of analytic techniques in combinatorial number theory for multimagic structures.

Abstract

In this paper we investigate $K$-multimagic squares of order $N$, these are $N \times N$ 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 $N_2(K)$ for which there exists nontrivial $K$-multimagic squares of order $N_2(K)$. Previous results on multimagic squares show that $N_2(K) \leqslant(4 K-2)^K$ for large $K$. Here we utilize the Hardy-Littlewood circle method and establish the bound $$ N_2(K) \leqslant 2 K(K+1)+1 $$ Via an argument of Granville's we additionally deduce the existence of infinitely many nontrivial prime valued $K$-multimagic squares of order $2 K(K+1)+1$.

Figures (2)

• Figure 1: Example of this partition represented via a coloring on $N \times N$ grid.
• Figure 2: Example of a bijection $\phi$ satisfying the above properties represented via labeling the $(i,j)$th entree on $N \times N$ grid the value $\phi^{-1}(i,j)$.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3