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Introduction to number theory. 2nd edition. (English) Zbl 1330.11001

Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-1749-6/hbk; 978-1-4987-1751-9/ebook). xii, 414 p. (2016).
The first edition of this textbook by the first two authors appeared in 2008, and was briefly reviewed back then in [Zbl 1138.11001]. The book under review is the largely reorganized second edition of this upper-level undergraduate primer on general number theory and its applications, this time with D. Garth as co-author replacing the meanwhile deceased Martin Erickson. As the authors point out in the preface to his new edition, the contents of the book have been rearranged to provide greater variety of options for the design of a one- or two-semester course by instructors, on the one hand, and to add some new material on the other hand. A further major change compared to the first edition concerns the numerous examples of computer-aided calculations and experimentations in number theory. Namely, in order to allow for greater flexibility for the instructor, these computational elements have been removed from the text and, instead, have been made available online, where they can be found at Vazzana’s web site tvazzana.sites.truman.edu/introduction-to-number-theory/. As a consequence of these significant changes, the current second edition comes with a finer subdivision into chapters (and their sections), but with more than hundred pages less, compared to the first edition of the book. More precisely, the eighteen chapters of the text under review cover the following topics: 6mm
1.
Introduction to natural numbers, integers, and the principle of mathematical induction.
2.
Divisibility properties and representations of integers.
3.
The greatest common divisor, the Euclidean algorithm, and linear Diophantine equations.
4.
Prime numbers and the fundamental theorem of arithmetic, with an outlook to the distribution of primes and Fermat’s Last Theorem.
5.
Linear congruences and the Chinese remainder theorem.
6.
Special congruences, including Fermat’s Little Theorem, Euler’s theorem, and Wilson’s theorem.
7.
Primitive roots and the construction of the regular 17-gon.
8.
Applications to cryptography.
9.
Quadratic congruences and the quadratic reciprocity law.
10.
Applications of quadratic residues: construction of tournaments and Hadamard matrices.
11.
Sums of squares and Gaussian integers.
12.
Further topics in Diophantine equations, including Fermat’s Last Theorem for the exponent 4, Pell’s equation, and the \(abc\) conjecture.
13.
Continued fractions and rational approximation of real numbers.
14.
Continued fraction expansions of quadratic irrationals and the solution of Pell’s equation.
15.
Arithmetic functions, including perfect numbers, Möbius inversion, partitions of an integer, and cyclotomic polynomials.
16.
Large primes, Fermat numbers, and Mersenne numbers.
17.
Analytic number theory: Chebyshev’s theorem, Bertrand’s postulate, the prime number theorem, the Riemann zeta function, and Dirichlet’s theorem on primes in arithmetic progressions.
18.
Elliptic curves and applications to encryption, factorization, and Fermat’s Last Theorem.
As one can see from the table of contents, and as the authors recommend in the preface, Chapters 1–7 should be covered as a solid foundation in an introductory course, while a selection of the other chapters (up to the instructor’s choice) could be used to complement the syllabus. Apart from the more advanced last two chapters, which require a higher level of the student’s mathematical maturity, the text is perfectly suited for upper-level undergraduates.
All together, the presentation of the material is self-contained, lucid, comprehensive, versatile and very inviting. The authors offer a broad spectrum of both classical and more recent topics in number theory, with numerous applications to other areas of mathematics as well as to daily-life problems, such as the ISBN system, the RSA encryption, or the construction of tournaments via arithmetic graph theory. The text is enhanced by a large number of illustrating examples and instructive exercises. In addition, each chapter comes with its own section of notes, in which historical remarks, biographies of towering figures in number theory, and other supplements are provided. No doubt, the reorganization of the material in this second edition of the book must be seen as a major improvement with regard to its utility, elegance, and reader-friendliness.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx Elementary number theory
11Dxx Diophantine equations
11G05 Elliptic curves over global fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11Uxx Connections of number theory and logic
11Y05 Factorization
11Y11 Primality
94A60 Cryptography

Citations:

Zbl 1138.11001
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