@inproceedings{936bb688fe6d4480bb14082b378472b7,

title = "Magic squares of squares over a finite field",

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{\textquoteright}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.",

keywords = "Magic square, Modulo arithmetics, Number theory",

author = "Stewart Hengeveld and Giancarlo Labruna and Aihua Li",

note = "Publisher Copyright: {\textcopyright} 2021 by the American Mathematical Society. All rights reserved.; 2nd International Meeting on Commutative Algebra and Related Areas, SIMCARA 2019, and the AMS Special Session on Commutative Algebra, 2019 ; Conference date: 14-09-2019 Through 15-09-2019",

year = "2021",

doi = "10.1090/conm/773/15536",

language = "English",

isbn = "9781470456016",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "111--122",

editor = "Baeth, {Nicholas R.} and Freitas, {Thiago H.} and Leuschke, {Graham J.} and {Jorge P{\'e}rez}, {Victor H.}",

booktitle = "Commutative Algebra",

address = "United States",

}