# MA257 Content

**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