Magic squares of squares over a finite field

Stewart Hengeveld, Giancarlo Labruna, Aihua Li

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

A magic square M over an integral domain D is a 3×3 matrix with entries from D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M are perfect squares in D, we call M a magic square of squares over D. In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z of the integers which has all the nine entries distinct?” We approach to answering a similar question when D is a finite field. We claim that for any odd prime p, a magic square over Zp can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p such that, over Zp, magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 (mod 120), there exist magic squares of squares over Zp that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

Original languageEnglish
Title of host publicationCommutative Algebra
Subtitle of host publication150 Years with Roger and Sylvia Wiegand
EditorsNicholas R. Baeth, Thiago H. Freitas, Graham J. Leuschke, Victor H. Jorge Pérez
PublisherAmerican Mathematical Society
Pages111-122
Number of pages12
ISBN (Print)9781470456016
DOIs
StatePublished - 2021
Event2nd International Meeting on Commutative Algebra and Related Areas, SIMCARA 2019, and the AMS Special Session on Commutative Algebra, 2019 - Madisson, United States
Duration: 14 Sep 201915 Sep 2019

Publication series

NameContemporary Mathematics
Volume773
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

Conference

Conference2nd International Meeting on Commutative Algebra and Related Areas, SIMCARA 2019, and the AMS Special Session on Commutative Algebra, 2019
Country/TerritoryUnited States
CityMadisson
Period14/09/1915/09/19

Keywords

  • Magic square
  • Modulo arithmetics
  • Number theory

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