# Constructing Magic Squares of Squares Modulo Certain Prime Numbers

5-2017

Thesis

## Degree Name

Master of Science (MS)

## College/School

College of Science and Mathematics

## Department/Program

Mathematical Sciences

Aihua Li

Mark S. Korlie

Bogdan G. Nita

## Abstract

A magic square is a square table of numbers such that each row, column, or diagonal adds up to the same sum. This research is inspired by an open question posed by Martin Labar in 1984. The open question states: “Can a 3 x 3 magic square be constructed using nine distinct perfect squares?” Though unsolved, this question sheds light on the existence of a Magic Square of Squares modulo a prime number p. For over two thousand years, many mathematicians have looked at these magical properties. In this thesis, the focus is on certain prime numbers p in the form of am + 1. We show that there exist Magic Squares of Squares with nine distinct elements mod p, for certain primes p. Constructions of such magic squares of squares are given. It is known that a magic square of squares can only admit 1, 2, 3, 5, 7, or 9 distinct numbers. We show that for infinitely many carefully selected prime numbers, non-trivial magic squares of squares with 2, 3, 5, 7, or 9 distinct perfect squares can be constructed. The results provide a positive answer to the open question regarding integers modulo certain prime numbers. The configurations used in the construction all have the appearance of 0, 1, 2, or 4. A further study investigates how many times each of these values can occur in a magic square of squares using the considered configurations. In addition, the constructions require the existence of quadruplet of consecutive quadratic residues. For each prime number considered, a set of such quadruplets is provided and used to construct desired magic squares of squares.

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