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What is Number Theory?

Unsurprisingly, number theorists are interested in the properties of numbers! In particular, the relations between the additive and multiplicative structures of integers are so fascinating that they make number theory a vast and fertile field of mathematical research. Gauss, who is often known as the 'prince of mathematics', called mathematics the 'queen of the sciences' and considered number theory the 'queen of mathematics'.

Many problems in number theory can be formulated in a relatively simple language. For example, the famous last theorem of Fermat states that there exists no non-trivial integer solution of the equation xn + yn = zn for n ≥ 3. The twin prime conjecture is to ask if there should be infinitely many primes p such that p+2 is also a prime. Although these questions are easily stated, they are so deep that they have withstood attempts to prove them for centuries. Fermat's last theorem was only proved in 1995 by A. Wiles after the cumulative effort of many mathematicians for more than 350 years, while the twin prime conjecture remains open even today.

The great difficulty in proving relatively simple results in number theory prompts many new concepts in mathematics. Over the centuries, this discipline has grown very much with close connections to other areas of mathematics: algebraic geometry, representation theory, combinatorics, cryptography and coding theory, probability, harmonic analysis and complex analysis.

The Pure Mathematics department teaches number theory courses at various levels. The elementary number theory course is offered twice per year. The analytic number theory and the algebraic number theory course are available every other year. We also teach advanced topic courses on a regular basis. Major areas of interest for number theorists in the Pure Mathematics department include Diophantine equations and Diophantine approximation, arithmetic geometry, L-functions and random matrix theory, representation theory, computational number theory, and additive and probabilistic number theory.