Award Date


Degree Type


Degree Name

Master of Science (MS)


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



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

x2 + y2 + z2 = 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 (x1,y1,z) and (x2,y2,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 = pr, where p is prime and also when z=kpr for p prime and k [mathematical equation refer to PDF]. In 2012, Chen proved the conjecture holds when 3z ± 2 = kpr 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 < 10140 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 < 10140 are proven to be unique. This is accomplished by writing computer algorithms to test each result.


Markov processes; Markov spectrum



File Format


Degree Grantor

University of Nevada, Las Vegas




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