A circle method approach to K-multimagic squares

IF 1.2 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-09-11 DOI:10.1112/jlms.70290

Daniel Flores

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引用次数: 0

Abstract

In this paper, we investigate $K$ -multimagic squares of order $N$ . These are $N \times N$ magic squares that 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 $N_{2} (K)$ for which there exist nontrivial $K$ -multimagic squares of order $N_{2} (K)$ . Previous results on multimagic squares show that $N_{2} (K) ⩽ {(4 K - 2)}^{K}$ for large $K$ . 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 $N ⩾ 4$ .

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k -多魔方的圆法求解

本文研究了K $K$ - N阶多魔方$N$。这些是N × N $N \times N$幻方，在将每个元素的k $k$次方提高到所有2±k±k之后，它们仍然是幻方$2 \leqslant k \leqslant K$。给定K小于2 $K \geqslant 2$，考虑存在非平凡K $K$ -多幻数的最小整数n2 (K) $N_2(K)$的建立问题n2 (K)阶的平方$N_2(K)$。先前在多魔方上的结果表明n2 (K)≥（4k−2）) K $N_2(K) \leqslant (4K-2)^K$为大K $K$。系数矩阵的复杂结构对圆法提出了重大的技术挑战。我们通过推广适用于圆形方法的丢番图系统类别并证明对于所有N或4 $N \geqslant 4$，多魔法正方形系统属于该类来克服这些障碍。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.