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Mathematics Institute

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 \Z_n . The multiplicative group U_n ; 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:

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

Objectives:

  1. 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.
  2. To stress the role of problem-solving in developing mathematical understanding.
  3. 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).
  4. 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.