On the existence of magic squares of powers
Rome N, Yamagishi S. 2025. On the existence of magic squares of powers. Research in Number Theory. 11(4), 91.
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Author
Rome, Nick;
Yamagishi, ShuntaroISTA
Corresponding author has ISTA affiliation
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Abstract
For any d 2, we prove that there exists an integer n0(d) such that there exists an n × n
magic square of dth 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 dth 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.
Publishing Year
Date Published
2025-09-23
Journal Title
Research in Number Theory
Publisher
Springer Nature
Acknowledgement
The authors are grateful to Tim Browning for his constant encouragement and enthusiasm, Jörg Brüdern for very helpful discussion regarding his paper [1] and Diyuan Wu for turning the proof of Theorem 2.4 in the original version into an algorithm and running the computation for us, for which the results are available in the appendix of the original version. They would also like to thank Christian Boyer for maintaining his website [4] which contains a comprehensive list of various magic squares discovered, Brady Haran and Tony Várilly-Alvarado for their public engagement activity of mathematics and magic squares of squares (A YouTube video “Magic Squares of Squares (are PROBABLY impossible)” of the Numberphile channel by Brady Haran, in which Tony Várilly-Alvarado appears as a guest speaker: https://www.youtube.com/watch?v=Kdsj84UdeYg.), and all the magic squares enthusiasts who have contributed to [4] which made this paper possible. Finally, the authors would like to thank the anonymous referees for their helpful comments, Daniel Flores for his work [11] which inspired them to optimise the proof of Theorem 2.4 and Trevor Wooley for very helpful discussion regarding recent developments in Waring’s problem and his comments on the original version of this paper.
Open access funding provided by Institute of Science and Technology (IST Austria). NR was supported by FWF project ESP 441-NBL while SY by a FWF grant (DOI 10.55776/P32428).
Volume
11
Issue
4
Article Number
91
eISSN
IST-REx-ID
Cite this
Rome N, Yamagishi S. On the existence of magic squares of powers. Research in Number Theory. 2025;11(4). doi:10.1007/s40993-025-00671-5
Rome, N., & Yamagishi, S. (2025). On the existence of magic squares of powers. Research in Number Theory. Springer Nature. https://doi.org/10.1007/s40993-025-00671-5
Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares of Powers.” Research in Number Theory. Springer Nature, 2025. https://doi.org/10.1007/s40993-025-00671-5.
N. Rome and S. Yamagishi, “On the existence of magic squares of powers,” Research in Number Theory, vol. 11, no. 4. Springer Nature, 2025.
Rome N, Yamagishi S. 2025. On the existence of magic squares of powers. Research in Number Theory. 11(4), 91.
Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares of Powers.” Research in Number Theory, vol. 11, no. 4, 91, Springer Nature, 2025, doi:10.1007/s40993-025-00671-5.
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