According to Gauss, ''Mathematics is the queen of the sciences and number theory is the queen of mathematics.'' Number theory can be divided, very roughly, into 3 parts:
Elementary number theory
Algebraic number theory
Analytic number theory
This course is concerned, basically, with elementary number theory, although we shall make a foray into the simplest topic in algebraic number theory, namely quadratic number fields. We shall also mention, without proof, the two basic results of analytic number theory, namely the Prime Number Theorem and Dirichlet's Theorem on primes in arithmetic sequences. We begin with the Fundamental Theorem of Arithmetic, Euclid's Theorem that every natural number n>0 is uniquely expressible as a product of primes. (This results is so familiar that one can easily overlook the subtlety of the proof, and the enormous step taken by Euclid or his school in establishing it.) Elementary number theory is, to a large extend, the study of prime numbers. As a kind of game to go with the course, we shall join the hunt for the next largest prime number. This is certain to be a Mersenne prime, due to the Lucas-Lehmer test which can tell us whether enormously large Mersenne numbers 2p 1(where p is a prime) are prime or not. We shall be able to establish, as an exercise in quadratic number fields, the validity of this test. On-line notes for the course are available in www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory In particular, the topics covered in the course can be found in the file Contents.pdf in that folder.
This module will be examined in a 2 hour examination in Trinity term. The final mark will be 80% exam plus 20% continuous assessment, consisting of 7 short problems each week. For supplementals, if required, the supplemental exam will count for 100%.