#### Date of Award

5-2018

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

#### Thesis Sponsor/Dissertation Chair/Project Chair

Aihua Li

#### Committee Member

Deepak Bal

#### Committee Member

Jonathan Cutler

#### Abstract

A magic square M over an integral domain D is a 3 x 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. Martin LaBar raised an open question in 1984, which states, “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 in case D is a finite field. Our main result confirms that a magic square of squares over a finite field F of characteristic greater than 3 can only hold 3, 5, 7, or 9 distinct entries. Corresponding to LaBar’s question, we claim that there are infinitely many prime numbers p such that, over a finite field of characteristic p, magic squares of squares with nine distinct elements exist. Constructively, we build magic squares of squares using consecutive quadratic residue triples derived from twin primes. We classify all the magic squares of squares over any finite fields of characteristic 2. Description of magic squares over a finite field of characteristic 3 is provided.

#### Recommended Citation

Labruna, Giancarlo, "Magic Squares of Squares of Order Three Over Finite Fields" (2018). *Theses, Dissertations and Culminating Projects*. 138.

https://digitalcommons.montclair.edu/etd/138