# MA3H1 Topics in Number Theory

Not Running in 2019/20

Lecturer:

Term(s):

Status for Mathematics students: List A

Commitment: 30 lectures, plus a willingness to work hard at the homework

Assessment: 15% by a number of assessed worksheets, 85% by 3-hour examination

Prerequisites: First-year mathematics and common sense. This module is independent of MA246 Number Theory and can be taken regardless of whether or not you have done MA246.

Leads To: MA3A6 Algebraic Number Theory, MA426 Elliptic Curves.

Content: We will cover the following topics:

1. Review of factorisation, divisibility, Euclidean Algorithm, Chinese Remainder Theorem.
2. Congruences. Structure on \$/m\$ and \$U_m\$. Theorems of Fermat and Euler. Primitive roots.
3. Quadratic reciprocity, Diophantine equations
4. Tonelli-Shanks, Fermat’s factorization, Quadratic Sieve.
5. Introduction to Cryptography (RSA, Diffie-Hellman)
6. p-adic numbers, Hasse Principle
7. Geometry of numbers, sum of two and four squares
8. Irrationality and trancendence
9. Binary quadratic forms, genus theory (ONLY if time allows!)

Books:

R. P. Burn, A Pathway into Number Theory, Cambridge University Press, 1997.

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.

I. Niven, H. S. Zukerman and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley, 1991.

H. E. Rose, A Course in Number Theory, Oxford University Press, 1988.

W. Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer-Verlag, 2008. Online version available from `http://modular.math.washington.edu/ent/ent.pdf`