Construction and Analysis of Magic Squares of Squares over Certain Finite Fields

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 \u3e 2. Furthermore, when p \u3e 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

Avoidability of long kk-abelian repetitions

We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are M\"akel\"a's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from M\"akel\"a. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary word

Constructing Magic Squares of Squares Modulo Certain Prime Numbers

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

Abelian-Square-Rich Words

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number of distinct abelian-square factors that is quadratic in $n$; and {\it uniformly abelian-square-rich} if every factor of $w$ contains a number of distinct abelian-square factors that is proportional to the square of its length. Of course, if a word is uniformly abelian-square-rich, then it is abelian-square-rich, but we show that the converse is not true in general. We prove that the Thue-Morse word is uniformly abelian-square-rich and that the function counting the number of distinct abelian-square factors of length $2n$ of the Thue-Morse word is $2$-regular. As for Sturmian words, we prove that a Sturmian word $s_{\alpha}$ of angle $\alpha$ is uniformly abelian-square-rich if and only if the irrational $\alpha$ has bounded partial quotients, that is, if and only if $s_{\alpha}$ has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the proof of Proposition

Clusters of repetition roots: single chains (Algebraic system, Logic, Language and Related Areas in Computer Sciences II)

This work proposes a new approach towards solving an over 20 years old conjecture regarding the maximum number of distinct squares that a word can contain. To this end we look at clusters of repetition roots, that is, the set of positions where the root u of a repetition u^[l] occurs. We lay the foundation of this theory by proving basic properties of these clusters and establishing upper bounds on the number of distinct squares when their roots form a chain with respect to the prefix order

Counting polyominoes: Yet another attack

A polyomino is a connected collection of squares on an unbounded chessboard. There is no known formula yielding the number of distinct polyominoes of a given number of squares. A polyomino enumeration method, faster than any previous, is presented. This method includes the calculation of the number of symmetric polyominoes. All polyominoes containing up to 24 squares have been enumerated (using ten months of computer time). Previously, only polyominoes up to size 18 were enumerated