Prime numbers: mysteries and records.
(Nombres premiers: mystères et records.)

*(French)*Zbl 0842.11001
Mathématiques. Paris: Presses Univ. de France. xx, 277 p. (1994).

Prime numbers have fascinated mathematicians, both professionals and amateurs, for a long time. The present book tries to present the history of the main achievements in answering the following questions:

1) How many primes are there?

2) How can one recognize if a number is prime or not?

3) Are there formulas or algorithms to describe all prime numbers?

4) How are the primes distributed among positive integers?

Each of the first four chapters is devoted to one of these questions. The fifth chapter presents some results about particular classes of prime numbers, while the last one deals with heuristical and probabilistic results about the primes. The book also contains references (general and for each chapter), a table of all prime numbers up to 10,000 and three indexes: one about the records presented in the books, one of the cited authors and a Subject Index. As the author explains in his conclusion, his intention was to write a not too big (but not too small) book about the mysteries and records of primes. Many of the proofs are omitted. The book offers the interested reader a nice story about the results of primes.

1) How many primes are there?

2) How can one recognize if a number is prime or not?

3) Are there formulas or algorithms to describe all prime numbers?

4) How are the primes distributed among positive integers?

Each of the first four chapters is devoted to one of these questions. The fifth chapter presents some results about particular classes of prime numbers, while the last one deals with heuristical and probabilistic results about the primes. The book also contains references (general and for each chapter), a table of all prime numbers up to 10,000 and three indexes: one about the records presented in the books, one of the cited authors and a Subject Index. As the author explains in his conclusion, his intention was to write a not too big (but not too small) book about the mysteries and records of primes. Many of the proofs are omitted. The book offers the interested reader a nice story about the results of primes.

Reviewer: C.Badea (Villeneuve d’Ascq)