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A concise introduction to the theory of numbers. (English) Zbl 0554.10001

Cambridge etc.: Cambridge University Press. XIII, 95 p. hbk: £15.00; $ 29.95; pbk: £4.95; $ 9.95 (1984).
This booklet is a concise but nearly complete introduction to number theory; as such it could be highly recommended. You can hardly imagine a topic of number theory which is not mentioned in it; at least an exercise or a reference in the commented bibliography at the end of each chapter is given.
A chapter ’Gauss and number theory’ serves as an introduction. His Disquisitiones arithmeticae are recommended as a general reference. The book opens with two chapters about divisibility and arithmetical functions. In the further chapters selected topics are fully discussed on a sophomoric level: I will mention, for example, the Chinese remainder theorem, a proof of the law of quadratic reciprocity, reduction of binary quadratic forms, continued fractions, a proof of the transcendence of \(e\), units in quadratic fields, and Euclidean quadratic fields.
The chapter on Diophantine equations contains discussions of Pell’s equation, Thue’s equation, Mordell’s equation \(y^ 2=x^ 3+k\) (with an outline of elliptic curves), and the Fermat equation (with proofs for exponent 3 and 4) and Catalan equation. Of course, the theme of linear forms in logarithms comes here again.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx Elementary number theory
11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
11D61 Exponential Diophantine equations
11J81 Transcendence (general theory)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11R11 Quadratic extensions
11R27 Units and factorization