# 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.