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Topics from the theory of numbers. (English) Zbl 0158.29501
New York: The Macmillan Company; London: Collier-Macmillan Ltd. 1966, xv, 299 p. 68 s. (1966).
This well-organized text is divided into three parts and two appendices. Part I is primarily a historical introduction. Part II consists of four chapters on elementary topics (divisibility, congruences, quadratic residues, and arithmetical functions). These two parts are accessible to any reader with a knowledge of elementary calculus. The most important part of the book is Part III, which contains three independent topics from analytic and algebraic number theory. Two chapters are devoted to the Riemann zeta function and an analytic proof of the prime number theorem. For this topic the required background in complex analysis is carefully outlined with copious references to the literature or to the appendix. – The next chapter deals with Fermat’s conjecture and goes as far as possible without the theory of ideals. This is followed by a chapter on ideal theory which is then used to prove the Fermat conjecture for regular primes in Case I. Case II is also treated in detail, with the exception of a theorem of Kummer which is stated without proof. This material lays an excellent foundation for the study of ideals in number fields and should serve to motivate students to pursue the subject further. – The last chapter returns to analytic number theory, specifically the theory of partitions. The use of generating functions is illustrated to derive Euler’s pentagonal number identity and related formulas. The chapter concludes with an elementary proof, following Erdős, of the inequality \(\exp\{(\alpha-\varepsilon)\sqrt{n}\}< p(n)< \exp\{\alpha\sqrt{n}\}\) for the partition function, where \(\alpha= \pi\sqrt{2/3}\) and \(\varepsilon>0\). The author writes with clarity and with contagious enthusiasm. His emphasis on the historical point of view gives the reader valuable insights and also shows how certain problems in the theory of numbers have influenced the development of other branches of mathematics. The book is a welcome contribution to the literature.
Reviewer: Tom M. Apostol

MSC:
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx Elementary number theory
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11N05 Distribution of primes
11N13 Primes in congruence classes
11Dxx Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11R11 Quadratic extensions
11R27 Units and factorization
11P81 Elementary theory of partitions