*(English)*Zbl 0856.11001

This monograph is the updated and enlarged 3rd edition of a well-known, informative, entertaining book, which can be read by number theorists with pleasure. The first edition (1988) was reviewed by H. C. Williams in Zbl 0642.10001. Due to the fast development in computational number theory, records given in the first edition are often out of date, and so, as the author writes in his elegant, witty style: “*The new book of prime number records* differs little from its predecessor in the general planning. But it contains new sections and updated records.

It has been comforting to learn about the countless computers (machines and men), grinding without stop, so that more lines with new large numbers could be added, bringing despair for the printers and proofreaders.”

In more detail, this monograph deals with elementary proofs for the infiniteness of the primes, with a 150-page-study of primality tests, with prime-representing functions (including “prime-producing” polynomials), with special primes (regular primes, Wieferich primes, “Sophie Germain primes”, all connected with “Fermat’s last theorem”; unfortunately E. Fouvry’s deep result on the first case of Fermat’s last theorem (“Sophie Germain primes”) is not mentioned), and with heuristic and probabilistic results on prime numbers.

The extensive bibliography covers more than 70 pages. Of course, it is not possible to describe in a short review the changes in the text improving the second edition (Example: the proof of $\pi \left(x\right)\to \infty $, using Fermat’s numbers ${F}_{n}={2}^{{2}^{n}}+1$, formerly attributed to Pólya, is, as the author points out, in fact due to Goldbach). The addenda make up for an enlargement of the size of the monograph by more than 10%. It ought to be mentioned that the printing of the 3rd edition is much nicer than that of the foregoing edition. Hopefully this monograph will induce many students to become interested in prime number theory.

##### MSC:

11-01 | Textbooks (number theory) |

11-02 | Research monographs (number theory) |

11A41 | Elementary prime number theory |

11N05 | Distribution of primes |

11B39 | Fibonacci and Lucas numbers, etc. |

11N13 | Primes in progressions |

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |

11P32 | Additive questions involving primes |

11Y11 | Primality |