On the existence of magic squares of powers

2024/06/13 by Nick Rome, Rome, Nick, Shuntaro Yamagishi +1
Computer Science · Physics and Astronomy · #05B15 #11D45 #11D72 #11G35 #11P55 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Graph Labeling and Dimension Problems #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.09364

openalex publication_date 2024/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any d ≥ 2, we prove that there exists an integer n₀(d) such that there exists an n × n magic square of d^th powers for all n ≥ n₀(d). In particular, we establish the existence of an n × n magic square of squares for all n ≥ 4, which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of n × n magic squares of d^th powers had only been known for sparse values of n. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity.

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