Progress in Mathematics (Boston, Mass.). 126. Boston, MA: Birkhäuser. xvi, 464 p. DM 158.00; öS 1232.40; sFr. 138.00; £ 69.00; FF 610.00 (1994).

Here is an outstanding technical monograph on recursive number theory and its numerous automated techniques. It successfully passes a critical milestone not allowed to many books, viz., a second edition. For a review of the first edition (1985) written by {\it H. J. te Riele} see

Zbl 0582.10001.
Many good things have happened to computational number theory during the ten years since the first edition appeared and the author includes their highlights in great depth. Several major sections have been rewritten and totally new sections have been added. The new material includes advances on applications of the elliptic curve method, uses of the number field sieve, and two new appendices on the basics of higher algebraic number fields and elliptic curves. Further, the table of prime factors of Fermat numbers has been significantly up-dated; e.g., now it is known that $F\sb 9$ is tri-composite and $F\sb{11}$ is penta-composite. Several other tables have been added so as to provide data to look for large prime factors of certain “generalized” Fermat numbers, while several other tables on special numbers were simply deleted in the second edition.
Still one can make several perplexing assertions or challenges: (1) prove that $F\sb 5$, $F\sb 6$, $F\sb 7$, $F\sb 8$ are the only four consecutive Fermat numbers which are bi-composite; (2) Show that $F\sb{14}$ is bi- composite. (This accounts for the difficulty in finding a prime factor for it.) (3) What is the smallest Fermat quadri-composite?; and (4) Does there exist a Fermat number with an arbitrarily prescribed number of prime factors?
All in all, this handy volume continues to be an attractive combination of number-theoretic precision, practicality, and theory with a rich blend of computer science.