Magic squares of squares over a finite field
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.
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Hengeveld, Stewart; Labruna, Giancarlo; and Li, Aihua, "Magic squares of squares over a finite field" (2021). Kean Publications. 1060.