Lectures on number theory. Ed. by Nikolaos Kritikos. Transl. from the German, with some additional material, by William C. Schulz.

*(English)*Zbl 0591.10001
Universitext. New York etc.: Springer-Verlag. xiv, 273 p. DM 58.00 (1986).

This is a textbook for a first course in elementary number theory. It consists of 6 chapters covering the usual material on divisibility, congruences, quadratic residues including reciprocity law, binary quadratic forms and continued fractions. Each chapter is followed by a set of problems compiled by the translator.

Considerable attention is paid to detail: if a proof is attempted it is done completely. And practically nothing is left for the reader, everything in the main stream of the book is dealt with quite exhaustively. Also, as a rule, numerical examples follow the definitions and theorems. Some outstanding results are quoted without proof and without any historical comment. For instance, the Prime Number Theorem is stated flatly “We have the following limiting relationships...” and the same is the fate of primes in arithmetic progressions. Among the very few names mentioned in the text are Gauss and Lagrange. Another, not so surprising, feature of the book is its complete independence from the language of algebra. Avoiding historical comments the book itself has an interesting history. It had never been seen, or – presumably – even planned by the author. A set of notes from a course in number theory given by Hurwitz at ETH in Zürich fell in hands of N. Kritikos who happened to be an enthusiastic student of Hurwitz in 1916–1917. He saved the material and now patronizes the edition of the book.

Considerable attention is paid to detail: if a proof is attempted it is done completely. And practically nothing is left for the reader, everything in the main stream of the book is dealt with quite exhaustively. Also, as a rule, numerical examples follow the definitions and theorems. Some outstanding results are quoted without proof and without any historical comment. For instance, the Prime Number Theorem is stated flatly “We have the following limiting relationships...” and the same is the fate of primes in arithmetic progressions. Among the very few names mentioned in the text are Gauss and Lagrange. Another, not so surprising, feature of the book is its complete independence from the language of algebra. Avoiding historical comments the book itself has an interesting history. It had never been seen, or – presumably – even planned by the author. A set of notes from a course in number theory given by Hurwitz at ETH in Zürich fell in hands of N. Kritikos who happened to be an enthusiastic student of Hurwitz in 1916–1917. He saved the material and now patronizes the edition of the book.

Reviewer: Kazimierz Szymiczek (Katowice)

##### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11Axx | Elementary number theory |

01A60 | History of mathematics in the 20th century |