Year Four


Number Theory
Course Code : NUMTH4
Units : 6
Lecturer : Delbrin Hussein Ahmed

Description :

  • Course overview:

Number theory is one of the oldest parts of mathematics. In its study of fundamental properties of numbers it uses every other part of mathematics and stimulates a variety of new developments in other areas. Number theory remains the most applicable part of pure mathematics through for example coding and cryptography and computer science.  Number theory concerns the solution of polynomial equations in whole numbers, or fractions. For example, the cubic equation x3 + y3 = z3 with x, y, z non-zero has infinitely many real solutions yet not a single solution in whole numbers. Equations of this sort are called Diophantine equations, and were first studied by the Greeks. What makes the study of these equations so fascinating is the seemingly chaotic distribution of prime numbers within the integers. We shall establish the basic properties of the Riemann zeta-function in order to find out how evenly these primes are distributed in nature. This module will present several methods to solve Diophantine equations including analytic methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems, and an introduction to more general modular forms.

  • Course objective:

This Course develops some of the main aspects of the theory of numbers with emphasis on Dirichlet series and their applications to the study of prime numbers. It builds on elementary number theory and uses techniques of complex analysis. This will provide the springboard for the study of more advanced topics which will be useful for further study of algebra, number theory and other areas of Mathematics. Students should also learn to present and develop a mathematical argument on a self-directed basis.