# Number theory

Number theory studies various properties of integers, from factorisation theory to distribution of prime numbers to finding integer solutions to polynomial equations. Its methods are based on intricate blends of algebraic and analytic arguments. Members of the Number Theory group are interested in computational, analytic, algebraic and geometric number theory and its application to pseudorandom number generation and dynamical systems.

### Group members

- Randell Heyman
- Mike Hirschhorn
- Bryce Kerr
- Alina Ostafe

- John Roberts
- Igor Shparlinski
- Tim Trudgian

(UNSW Canberra) - Liangyi Zhao

The group also has strong ties with the Number Theory group at UNSW Canberra.

### Research interests

**David Angell**

He is interested in number theory and combinatorics, particularly continued fractions, irrationality and transcendence. A selection of extension articles for secondary students can be found at his personal homepage. The image at the top of the page illustrates an exponential sum involving the function f(n)=n/dd+n^2/mm+n^3/yy, where dd.mm.19yy is David's birthday.

**Peter Brown**

He has been working in the area of number theory, specifically on elliptic curves and has recently been looking at some problems in analytic number theory.

**David Harvey **

His research interests lie in the areas of computational number theory, polynomial and integer arithmetic, and arithmetic geometry.

**Randell Heyman**

His research interests include arithmetic functions, irreducible polynomials, pairwise coprimality and finite fields.

**Bryce Kerr**

Interested in additive combinatorics, exponential sums and character sums.

**Mike Hirschhorn **

He studies applications of q-series to problems in additive number theory. A greater part of his work is bound up in elucidating results due to Ramanujan.

**Alina Ostafe**

Interested in algebra and number theory, particularly in algebraic dynamical systems, polynomials and rational functions over finite fields and their applications to pseudorandom number generators and cryptography. In her research, she uses various tools of analytic number theory (exponential and character sums, additive combinatorics) and commutative algebra (discriminants, resultants, Hilbert’s Nullstellensatz).

**John Roberts**

His research is in dynamical systems (sometimes popularly termed "Chaos Theory"), which seeks to understand how systems change with time and how this evolution can be understood, classified and predicted. This area of research is an exciting interdisciplinary field which relates to, and uses ideas from pure and applied mathematics, physics and computer science. His current work focuses on two broad areas: the study of integrable systems (ordered dynamics based on rotations) and the study of arithmetic dynamics (a hybrid of dynamical systems with number theory).

**Igor Shparlinski **

He has a broad range of research interests, from number theory (such as exponential and character sums, finite fields, smooth numbers, linear recurrence sequences), cryptography (especially elliptic curve cryptography and pseudorandom number generators) and computational aspects including algorithm design, computational complexity and quantum cryptography.

**Tim Trudgian**

He is interested in classical analytic number theory, including the properties of the Riemann zeta-function, Dirichlet L-functions, and their applications to the distribution of primes and primitive roots.

**Liangyi Zhao**

A researcher analytic number theory. More specifically, he is interested in studying exponential sums, character sums, L-functions and the distribution of prime numbers in special sequences.