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.
- 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.
- 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.
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.