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MA257 Introduction to Number Theory

Lecturer: Hong Liu

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:

Useful background: Interest in Number Theory is essential.

Synergies:

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
  • 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.

Additional Resources