MT311 Number Theory (Term 1: Dr E J Scourfield)
Prerequisite: MT181
Teaching: 33hr lectures
Assessment: 2 hr written examination
Aims
To acquaint students with some of the elementary tools used to analyse the additive and multiplicative structures of the set of integers.
Learning outcomes
On completion of this course students should:
· Be confident in handling congruences, including the use of the Chinese Remainder theorem, and the Fermat-Euler theorem;
·
Be able to manipulate arithmetic functions such
as
and 
and derive some of
their basic properties;
· Be able to prove the existence of primitive roots modulo a prime and use them in solving certain congruences;
· Be able to test for quadratic residues, and use them to answer questions on primes in arithmetic progressions, and representing numbers as sums of two squares;
· Be able to find the continued fraction expansion of real numbers, in particular quadratic irrationals, and apply continued fractions to the solution of Pell’s equation.
Syllabus
Introduction. Revision of material seen in previous years. Integers, primes, factorization, congruences including Chinese remainder theorem and the Fermat-Euler theorem.
Arithmetic functions. Introduction to the
functions 
and 
product and summation
form, Mobius inversion.
Primitive roots. The order of an integer modulo p, primitive roots modulo p, the proof of their existence, the index of an integer modulo p.
Quadratic residues. Legendre’s symbol. Euler’s criterion for a quadratic residue. Gauss’s Lemma. The law of quadratic reciprocity. Applications to quadratic congruences and primes in arithmetic progressions.
Continued fractions. Definition. Diophantine approximation. Quadratic irrationals and Pell’s equation.
Arithmetic functions II. The function r(n) – representing numbers as sums of two squares.
Indicative Texts
Introduction to the Theory of Numbers – I
Niven, H S Zuckerman and H L
Elementary Number Theory in Nine Chapters – J J
Tattersall. (