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