# 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:**

- MA136 Introduction to Abstract Algebra: Rings, subrings, ideals, integral domains, fields
- MA132 Foundations or MA138 Sets and Numbers: Congruence modulo n, prime factorisation, Euclidean algorithm, gcd and lcm, Bezout Lemma

**Useful**** background:** Interest in Number Theory is essential

**Synergies: **

- MA249 Algebra II: Groups and Rings: Ring theoretic part Algebra II and Introduction to Number Theory have much in common

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

- MA3G6 Commutative Algebra
- MA3A6 Algebraic Number Theory
- MA4H9 Modular Forms
- MA4L6 Analytic Number Theory
- MA426 Elliptic Curves
- MA4H8 Ring Theory

**Content:**

- Factorisation, divisibility, Euclidean Algorithm, Chinese Remainder Theorem
- Congruences. Structure on Z/mZ and its multiplicative group. Theorems of Fermat and Euler. Primitive roots
- Quadratic reciprocity, Diophantine equations
- Elementary factorization algorithms
- Introduction to Cryptography
- p-adic numbers, Hasse Principle
- 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