#### Degree Name

Master of Arts (MA)

#### Semester of Degree Completion

2016

#### Thesis Director

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*+*bω* ∣ *a*, *b* ∈ ℤ} where *ω*^{2} + *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* + *bi* ∣ *a*, *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.

#### Recommended Citation

Bandara, Sarada, "An Exposition of the Eisenstein Integers" (2016). *Masters Theses*. 2467.

https://thekeep.eiu.edu/theses/2467