Hardy G.H., Wright E.M. An Introduction to the Theory of Numbers

Oxford University Press, 1971. - 421 pages.This book has developed gradually from lectures delivered in a number of universities during the last ten years, and, like many books which have grown out of lectures, it has no very definite plan.It is not in any sense (as an expert can see by reading the table of contents) a systematic treatise on the theory of numbers. It does not even contain a fully reasoned account of any one side of that many-sided theory, but is an introduction, or a series of introductions, to almost all of these sides in turn. We say something about each of a number of subjects which are not usually combined in a single volume, and about some which are not always regarded as forming part of the theory of numbers at all. Thus Chs. XII-XV belong to the 'algebraic' theory of numbers, Chs. XIX-XXI to the 'additive', and Ch. XXII to the 'analytic' theories; while Chs. Ill, XI, XXIII, and XXIV deal with matters usually classified under the headings of 'geometry of numbers' or 'Diophantine approximation. There is plenty of variety in our programme, but very little depth; it is impossible, in 400 pages, to treat any of these many topics at all profoundly.There are large gaps in the book which will be noticed at once by any expert. The most conspicuous is the omission of any account of the theory of quadratic forms. This theory has been developed more systematically than any other part of the theory of numbers, and there are good discussions of it in easily accessible books. We had to omit something, and this seemed to us the part of the theory where we had the least to add to existing accounts.The book is written for mathematicians, but it does not demand anv great mathematical knowledge or technique. In the first eighteen chapters we assume nothing that is not commonly taught in schools, and any intelligent university student should find them comparatively easy reading. The last six are more difficult, and in them we presuppose