#### 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

Baojun Song

#### Committee Member

Bogdan G. Nita

#### Subject(s)

Magic squares

#### Abstract

This work is dedicated to the properties of the 3 × 3 magic squares of cubes modulo a prime number. Its central concept is the number of distinct entries of these squares and the properties associated with this number. We call this number the degree of a magic square. The necessary conditions for the magic square of cubes with degrees 3, 5, 7, and 9 are examined. It was established that there are infinitely many primes for which magic squares of cubes with degrees 3, 5, 7, and 9 exist. I apply n-tuples of consecutive cubic residues to prove that there are infinitely many Magic Squares of Cubes with degree 9. Furthermore I use Brauer’s theorem, that guarantees the existence of a sequence of consecutive integers of any length, to construct Magic Squares of Cubes whose entries are all cubic residues. Such analytic tools as Modular Arithmetic, Legendre symbol, Fermat’s Little Theorem, notions of quadratic and cubic residues were employed in the process of research.

#### Recommended Citation

Sokolovskiy, Yevgeniy, "Magic Squares of Cubes Modulo a Prime Number" (2018). *Theses, Dissertations and Culminating Projects*. 152.

https://digitalcommons.montclair.edu/etd/152