# MA246 Content

**Content**:

*Workbook 1: Modular arithmetic: congruences.* The arithmetic of congruence classes; solving linear congruences and linear Diophantine Equations via the extended Euclidean Algorithm.

*Workbook 2: Modular arithmetic: the ring ** .* The multiplicative group ; Euler's phi function; Euler's Theorem and Fermat's Little Theorem; the Chinese Remainder theorem.

*Workbook 3: Primitive roots and discrete logarithms.* Existence (and number) of primitive roots modulo *p*; existence and non-existence of primitive roots mod *n*. Finite logarithms.

*Workbook 4: Continued fractions I.* Continued fractions: finite expansions for rational numbers, infinite expansion for irrational numbers, periodic expansions for quadratic irrationals.

*Workbook 5: Continued fractions II.* Periodic continued fraction expansions for quadratic irrationals (concluded). Application to Pell's equation.

**Aims**:

- To introduce students to the delights of elementary number theory.
- To encourage independent study through using specially-prepared workbooks which develop abstract theory through sequences of concrete exercises, problems, and calculations.

**Objectives**:

- To give students an easy facility with modular arithmetic, continued fractions, and the elementary functions of number theory; in particular, to develop their ability to do serious calculations with these objects.
- To stress the role of problem-solving in developing mathematical understanding.
- To stimulate the use of calculations in investigative mathematics (but note that calculators will not be needed and will not be allowed in the examination).
- To provide an incentive for cooperative study.

**Books**:

Prices are from `amazon.co.uk`

(2007) where available.

**First Choice:** H. Davenport, *The Higher Arithmetic*, 7th Edn. (CUP, 1999), ISBN 0521634466, £22.79.

**A Problems-Based Approach** R P. Burn, *A Pathway into Number Theory* (CUP, 2nd edition, 1997), ISBN 0521575400, £22.79.

**Also Recommended:** J. H. Silverman, *A Friendly Introduction to Number Theory* (Prentice-Hall, 2nd edition, 2001), ISBN 0130309540, used and new from £18.98.

**Also Recommended:** J. J. Tattersall, *Elementary Number Theory in Nine Chapters* (CUP, 1999), ISBN 0521585317, £18.99.

**The Classic:** G. H. Hardy & E. M. Wright, *An Introduction to the Theory of Numbers* (OUP, 1980), ISBN 0198531710, used and new from £17.81.

**A Number Theorist's World View:** G. H. Hardy, *A Mathematician's Apology* (CUP, 1992), ISBN 0521427061, £8.59.