Elementary number theory (Q2460698)

The first edition of this popular primer of elementary number theory was published twenty years ago (Zbl 0622.10001). Based upon an introductory course that R. Remmert has taught repeatedly in the past, this text covers the material of elementary number theory which is fundamental and indispensable for every student, instructor, school teacher, and interested autodidact in mathematics. Keeping the abstract prerequisites to a minimum, the authors provide a very profound and cultured introduction to the main topics of the subject, which is reflected by the seven chapters of the book. Chapter 1 is devoted to the very basics concerning the concept of divisibility for integers, prime numbers, the prime decomposition in \(\mathbb{Z}\) and \(\mathbb{Q}\), perfect numbers, Fermat and Mersenne primes, and irrationality problems for real numbers. Chapter 2 develops the theory of greatest common divisors in \(\mathbb{Z}\), their ideal-theoretic characterization, and their application to the study of both the representation and distribution of prime numbers, including a discussion of big primes, Chebyshev's elementary estimate for \(\pi(x)\), primes in arithmetic progressions, and primes as values of certain polynomials. Chapter 3 turns to the more general theory of divisibilty in integral domains, with the focus on principal ideal domains, factorial rings, and Euclidean rings. Chapter 4 treats \(g\)-adic algorithms in \(\mathbb{Z}\) and \(\mathbb{Q}\), whereas Chapter 5 returns to the more algebraic aspects by explaining linear congruences, the factor rings \(\mathbb{Z}/n\mathbb{Z}\), and the classical theorems of Fermat-Euler, Wilson, Lagrange, along with the Chinese remainder theorem. Chapter 6 introduces the basic concepts of elementary group theory and, in the sequel, studies the structure of the groups of multiplicative units in the factor rings \(\mathbb{Z}/n\mathbb{Z}\). Chapter 7, the final chapter of the book, discusses quadratic congruences, the Legendre symbol, the Jacobi symbol, and -- as the point of culmination of the entire text -- the jewel of elementary number theory: Gauss's fundamental quadratic law of reciprocity and its supplements. Apart from the classical proof using the famous Gauss's Lemma, the fascinating analytical proof by Eisenstein (1845) is presented as well. All together, the whole exposition stands out by its unrivalled lucidity, its user-friendly comprehensiveness, its unique wealth of historical and philological comments, and its rich supply of accompanying exercises to each single section of the book. The entire text is absolutely self-contained, although it requires some solid familiarity with the contemporary language and terminology of mathematics based on set theory. The current third edition of this well-tried introduction to elementary number theory is basically identical with the original (Zbl 0622.10001) and its (slightly corrected) second edition (Zbl 0824.11002) from 1995. However, the authors have undertaken a few further polishing revisions and some minor completions of the material, thereby enhancing the value of this German standard text once more. No doubt, this excellent primer of elementary number theory deserves a wider, international audience of readers by means of a translation into English, on the occasion of which an appendix providing solutions to selected exercises for less experienced readers possibly could be added.