Magic Squares of Squares of Order 5 Modulo a Prime Number

In this paper, I examine magic squares of squares (MSS) of order 5 over Zp where p is a prime number. The first approach to the problem is to find how many distinct elements an MSS may have (called the degree of the MSS). In the next step, I study the relationship between the magic sum and the center entry of any MSS. In order to develop construction methods and configurations for magic squares of squares of order 5 with desired degrees, I study Pythagorean triples and sequences of consecutive quadratic residues modulo p. Properties of these sequences are provided and applied to construct desired magic squares of squares. This research focuses on magic squares of squares of order 5 in which the center 3 x 3 square is a magic square of squares of order 3. I claim that the magic sum of such an MSS M is 5c, where c is the center element of M and the degree of M must be odd when p > 5. The main results of the thesis include several configurations for the construction of MSS of a given degree and the existence ofMSSs of all possible odd degrees over Zp for infinitely many primes p. Chapter 1 presents an overview of modular arithmetic as well as some important definitions. Chapter 2 gives the results about the magic sum and degrees. In Chapter 3, I investigate special sequences of quadratic residues and describe properties of them. In Chapter 4, by applying special sequences of quadratic residues, several configurations are developed and they are used to construct MSSs of a given degree. The main results of this thesis are provided in Chapter 4 as well.