---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20423'
abstract:
- lang: eng
  text: "For any d  2, we prove that there exists an integer n0(d) such that there
    exists an n × n\r\nmagic square of dth powers for all n  n0(d). In particular,
    we establish the existence of\r\nan n × n magic square of squares for all n  4,
    which settles a conjecture of\r\nVárilly-Alvarado. All previous approaches had
    been based on constructive methods and\r\nthe existence of n × n magic squares
    of dth powers had only been known for sparse\r\nvalues of n. We prove our result
    by the Hardy-Littlewood circle method, which in this\r\nsetting essentially reduces
    the problem to finding a sufficient number of disjoint linearly\r\nindependent
    subsets of the columns of the coefficient matrix of the equations defining\r\nmagic
    squares. We prove an optimal (up to a constant) lower bound for this quantity."
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.\r\nOpen 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)."
article_number: '91'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Nick
  full_name: Rome, Nick
  last_name: Rome
- first_name: Shuntaro
  full_name: Yamagishi, Shuntaro
  id: 0c3fbc5c-f7a6-11ec-8d70-9485e75b416b
  last_name: Yamagishi
citation:
  ama: Rome N, Yamagishi S. On the existence of magic squares of powers. <i>Research
    in Number Theory</i>. 2025;11(4). doi:<a href="https://doi.org/10.1007/s40993-025-00671-5">10.1007/s40993-025-00671-5</a>
  apa: Rome, N., &#38; Yamagishi, S. (2025). On the existence of magic squares of
    powers. <i>Research in Number Theory</i>. Springer Nature. <a href="https://doi.org/10.1007/s40993-025-00671-5">https://doi.org/10.1007/s40993-025-00671-5</a>
  chicago: Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares
    of Powers.” <i>Research in Number Theory</i>. Springer Nature, 2025. <a href="https://doi.org/10.1007/s40993-025-00671-5">https://doi.org/10.1007/s40993-025-00671-5</a>.
  ieee: N. Rome and S. Yamagishi, “On the existence of magic squares of powers,” <i>Research
    in Number Theory</i>, vol. 11, no. 4. Springer Nature, 2025.
  ista: Rome N, Yamagishi S. 2025. On the existence of magic squares of powers. Research
    in Number Theory. 11(4), 91.
  mla: Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares of Powers.”
    <i>Research in Number Theory</i>, vol. 11, no. 4, 91, Springer Nature, 2025, doi:<a
    href="https://doi.org/10.1007/s40993-025-00671-5">10.1007/s40993-025-00671-5</a>.
  short: N. Rome, S. Yamagishi, Research in Number Theory 11 (2025).
corr_author: '1'
das_tickbox: '0'
date_created: 2025-10-05T22:01:34Z
date_published: 2025-09-23T00:00:00Z
date_updated: 2026-07-17T12:16:15Z
day: '23'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s40993-025-00671-5
external_id:
  arxiv:
  - '2406.09364'
file:
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  creator: dernst
  date_created: 2025-10-13T11:28:49Z
  date_updated: 2025-10-13T11:28:49Z
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has_accepted_license: '1'
intvolume: '        11'
issue: '4'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P32428
  name: New frontiers of the Manin conjecture
publication: Research in Number Theory
publication_identifier:
  eissn:
  - 2363-9555
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: On the existence of magic squares of powers
tmp:
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  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 11
year: '2025'
...