## Theses, Dissertations and Culminating Projects

5-2012

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

Aihua Li

Jonathan Cutler

#### Committee Member

William Parzynski

#### Abstract

This thesis presents results of the research on the study of Magic Squares of Squares over certain finite fields. The research is motivated by an open question, which is still not answered: "Does there exist a 3 x 3 magic square with all nine entries being distinct perfect squares of integers?" Instead of directly trying to answer this challenging question, this research attempts to answer a parallel question: "Does there exist a 3 x 3 magic square with all nine entries being distinct perfect squares modulo a prime number p?" Equivalently, the question can be restated as, "Does there exist a 3 x 3 magic square with all nine entries being distinct quadratic residues of a prime number p?" It is shown in this thesis that the answer is "Yes" for some primes such as 29 and 59, but "No" for many other primes like 17 and 19.

Consider a prime number p and the finite field Zp. The focus of this research is on the existence, analysis, and construction of the magic squares of squares made of quadratic residues of p from Zp. The main results show that such a magic square of squares can only use an odd number of distinct quadratic residues of p when p > 2. Furthermore, when p > 3, there exist magic squares of squares over Zp with 3 distinct entries. When p = I mod 8, there exist magic squares of squares over Zp made of five distinct quadratic residues of p. Existence of magic squares of squares over Zp made of seven or nine distinct numbers is also discussed. Investigation has been done toward answering the question: "What is the maximum number of distinct quadratic residues of p that a magic square of squares over Zp can admit?"

Chapter 6 of the thesis contains results from a related educational project. Through an MSU GK-12 program funded by NSF (Award #0638708), the author introduced magic squares and the mathematics involved in finding them to middle school students as a part of the project "Integrating Graduate Research into Middle School Class­ rooms". The findings are given in this chapter.

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