#### Award Date

5-1-2015

#### Degree Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematical Sciences

#### First Committee Member

Arthur Baragar

#### Second Committee Member

Ebrahim Salehi

#### Third Committee Member

Peter Shiue

#### Fourth Committee Member

Stephen Miller

#### Number of Pages

49

#### Abstract

In this thesis we discuss the positive integer solutions to the equation known as the Markoff equation

x^{2} + y^{2} + z^{2} = 3xyz.

Each solution to the equation is a permutation of a triple (x,y,z) with [mathematical equation refer to PDF] which is called a Markoff triple and each integer of the triple is referred to as a Markoff number.

In 1913, Frobenius conjectured that given an ordered Markoff triple (x,y,z), then both x and y are uniquely determined by z. In other words, if both (x_{1},y_{1},z) and (x_{2},y_{2},z) are solutions to the Markoff equation with [mathematical equation refer to PDF]. When this is true for a particular z, we say that z is unique. Since the time of Frobenius there have been numerous results on what we refer to now as the Frobenius Conjecture.

In 1996 Baragar proved that given a Markoff number z, it is unique whenever z, 3z-2, or 3z+2 is a prime, twice a prime or four times a prime. In 2001, Button proved that z is unique whenever z = p^{r}, where p is prime and also when z=kp^{r} for p prime and k [mathematical equation refer to PDF]. In 2012, Chen proved the conjecture holds when 3z ± 2 = kp^{r} for p prime and k [mathematical equation refer to PDF]. There is a recent result due to Srinivasan that utilizes divisors of the discriminant of quadratic forms, the details of which will be explained in the thesis.

The goal of this thesis is to empirically investigate how “good” these results are, in the sense that we wish to know how many Markoff triples are shown to be unique with each successive result. In Baragar’s paper from 1996, it was shown that all Markoff triples with z < 10^{140} are unique, and that approximately 6% of them satisfied the conditions of his main result. Due to the results from Button (2001) and Chen (2012), roughly 60% of all Markoff triples with z < 10^{140} are proven to be unique. This is accomplished by writing computer algorithms to test each result.

#### Keywords

Markov processes; Markov spectrum

#### Disciplines

Mathematics

#### Language

English

#### Repository Citation

Metz, Brandon John, "A Comparison of Recent Results on the Unicity Conjecture of the Markoff Equation" (2015). *UNLV Theses, Dissertations, Professional Papers, and Capstones*. 2389.

http://digitalscholarship.unlv.edu/thesesdissertations/2389