Number Theory

Module code: MA3153

Number theory is one of the oldest branches of pure mathematics, and one of the largest. It concerns questions about numbers, usually meaning integers or rational numbers (fractions). 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.

In order to study integers, it is necessary to understand the atoms of integers -prime numbers. Every integer can be written as a product of prime numbers. Many questions and problems in number theory are therefore directly related to questions and properties of prime numbers. For example, the famous Goldbach Conjecture asks if every even integer greater than 3 can be written as the sum of two primes. It is easy to check examples such as 10=3+7, 12=5+7,  but so far no proof is known and it is one of the oldest unsolved problems in mathematics. Prime numbers also play an essential role in cryptography and coding theory and therefore are very important for data security.

In number theory to solve problems about integers, which sometimes can be understood without much prior knowledge of mathematics as the two examples above show, require very often the use of tools and methods from a vast area of different mathematical disciplines and in this module, we will see the use of some of them, mainly from algebra and analysis. A main part of this module will be devoted to the study of prime numbers using algebraic and analytic concepts.