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#### Multiplicities of Linear Recurrence Sequences

(University of Waterloo, 2006)

In this report we give an overview of some of the major results concerning the multiplicities of linear recurrence sequences. We first investigate binary recurrence sequences where we exhibit a result due to Beukers and a result due to Brindza, Pintér and Schmidt. We then investigate ternary recurrences and exhibit a result due to Beukers building on work of Beukers and Tijdeman. The last two chapters deal with a very important result due to Schmidt in which we bound the zero-multiplicity of a linear recurrence sequence of order

**t**by a function involving**t**alone. Moreover we improve on Schmidt's bound by making some minor changes to his argument....#### Generalisations of Roth's theorem on finite abelian groups

(University of Waterloo, 2012-12-18)

Roth's theorem, proved by Roth in 1953, states that when A is a subset of the integers [1,N] with A dense enough, A has a three term arithmetic progression (3-AP). Since then the bound originally given by Roth has been ...

#### On Transcendence of Irrationals with Non-eventually Periodic b-adic Expansions

(University of Waterloo, 2010-05-18)

It is known that almost all numbers are transcendental in the sense of Lebesgue measure. However there is no simple rule to separate transcendental numbers from algebraic numbers. Today research in this direction is about ...

#### The Prouhet-Tarry-Escott problem

(University of Waterloo, 2013-01-15)

Given natural numbers n and k, with n>k, the
Prouhet-Tarry-Escott (PTE) problem asks for distinct
subsets of Z, say X={x_1,...,x_n} and
Y={y_1,...,y_n}, such that
x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. ...

#### On a Question of Wintner Concerning the Sequence of Integers Composed of Primes from a Given Set

(University of Waterloo, 2007-09-27)

We answer to a Wintner's question
concerning the sequence of integers
composed of primes from a given set.
The results generalize and develop the answer to Wintner’s question due to
Tijdeman.

#### Equality of Number-Theoretic Functions over Consecutive Integers

(University of Waterloo, 2009-04-30)

This thesis will survey a group of problems related to certain number-theoretic functions. In particular, for said functions, these problems take the form of when and how often they are equal over consecutive integers, n ...