# MA257 Introduction to Number Theory

Lecturer: Sam Chow

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 one hour lectures

Assessment: 85% 2 hour examination, 15% homework assignments

Formal registration prerequisites: None

Assumed knowledge:

• Ring theory: rings, subrings, ideals, integral domains, fields
• Number theory: congruence modulo n, prime factorisation, Euclidean algorithm, gcd and lcm, Bezout Lemma

Useful background: Interest in Number Theory is essential

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content:

• Factorisation, divisibility, Euclidean Algorithm, Chinese Remainder Theorem
• Congruences. Structure on Z/mZ and its multiplicative group. Theorems of Fermat and Euler. Primitive roots
• Elementary factorization algorithms
• Introduction to Cryptography
• Geometry of numbers, sum of two and four squares

Aims: To introduce students to elementary number theory and provide a firm foundation for later number theory and algebra modules

Objectives: By the end of the module the student should be able to:

• Work with prime factorisations of integers
• Solve congruence conditions on integers
• Determine whether an integer is a quadratic residue modulo another integer
• Apply p-adic and geometry of numbers methods to solve some Diophantine equations
• Follow advanced courses on number theory in the third and fourth year

Books:
H. Davenport, The Higher Arithmetic, Cambridge University Press

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1979

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1990