Document Type

Thesis - Open Access

Award Date


Degree Name

Master of Science (MS)


Mathematics and Statistics


History tells us that one of the most interesting topics of mathematics is elementary number theory, the arithmetic Gauss spoke of when he said, “Mathematics is the queen of the sciences and arithmetic the queen of mathematics." We will investigate one conjecture concerning the theory of numbers. Throughout this paper small case Latin letters with the exception of e and i will represent integers where the set of rational integers is denoted by Z. 1he Latin letters e and i respectively represent the base for the natural logarithms and the so-called imaginary unit for the set of complex numbers. An Eisenstein integer, a, is a complex number that can be written as a = a + bw where a and b are rational integers and w is the cube root of unity, e2wi/3, so that w = (-1 +i/3)/2. We denote the set of Eisenstein integers by Z(w) = {a + bw ׀ a,b ε Z} and let the Greek letters a, ꞵ, ү, p, and δ represent integers in Z(w). Since w is a cube root of unity, w2 + w + 1 = O. The set of integers in Z(w) will be represented geometrically by the lattice points in a Cartesian coordinate system formed by the intersections of the lines through the points (m,0) and making angles of 60° or 120° with the x-axis. The system is a honeycomb of equilateral triangles.

Library of Congress Subject Headings

Number theory



Number of Pages



South Dakota State University