Date of Award

2016

Degree Type

Thesis

Degree Name

Master of Arts (MA)

Author's Department

Mathematics and Computer Science

First Advisor

Alejandra Alvarado

Abstract

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves.

An example of said ring are complex numbers of the form ℤ[ω] = {a+a, b ∈ ℤ} where ω2 + ω + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[i] = {a + bia, b ∈ ℤ} which form a square lattice in the complex plane. The Gaussian moat problem, first posed by Basil Gordon in 1962 at the International Congress of Mathematicians in Stockholm [7], asks whether it is possible to "walk" from the origin to infinity using the Gaussian primes as "stepping stones" and taking steps of bounded length.

Although it has been shown that one cannot walk to infinity on the real number line, taking steps of bounded length and stepping only on the primes, the moat problem for Gaussian and Eisenstein primes remains unsolved. We will provide the necessary background for the reader, then investigate the Eisenstein moat problem.

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

Included in

Number Theory Commons

Share

COinS