Factorizations of bn + 1 : b=2,3,5,6,7,10,11,12 up to high powers / John Brillhart, D. H. Lehmer, J. L. Selfridge, ... [et al]Type de document : MonographieCollection : Contemporary mathematics, 22Langue : anglais.Pays : Etats Unis.Éditeur : Providence : American Mathematical Society, 1983Description : 1 vol. (LXVII-178 p.) : tables, appendix ; 26 cmISBN : 9780821891209.ISSN : 0271-4132.Bibliographie : Notes bibliogr..Sujet MSC : 11-04, Number theory, Explicit machine computation and programs (not the theory of computation or programming)
11A41, Number theory -- Elementary number theory, Primes
11B39, Number theory -- Sequences and sets, Fibonacci and Lucas numbers and polynomials and generalizations
65A05, Numerical analysis -- Tables, Tables
68Wxx, Computer science, AlgorithmsEn-ligne : Zentralblatt | MathSciNet
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The bulk of the body of this book consists of the tables of factorings mentioned in the title. The "very high powers'' range from 135 for powers of 12±1 to as high as 2400 for powers of 2 in which the exponent is of the form 4k−2. The format of the tables takes a little getting used to. Long prime (and probably prime) factors and cofactors known to be composite but as yet not factored are placed in appendices at the end so as not to interfere unduly with the physical layout of the main tables. In addition, "obvious'' algebraic factorings are indicated symbolically, with the earlier text explaining the algebra. In spite of the cross-references and notation, the tables can be read easily with only a little practice, and certainly carry far more meaning than they would if the algebraic factorings were simply presented explicitly.
The very long (68 pages) introduction contains, in addition to a set of short tables of factorings of powers of 2 and of 10 (plus and minus 1), a history and a mathematical history of the original Cunningham-Woodall tables, the developments in the theory and practice of factoring and of primality testing, the not-unrelated developments in computer technology and computing machines since the publication of the original tables, a careful explanation of the "obvious'' algebraic factoring of those integers for which such a factoring exists, and a status report on the ongoing work to finish the factoring of cofactors and to prove the primality of the probably-prime factors.
The third part of this book does not appear here. It exists as thousands of lines of code stored electronically in many different computers, or once existed as miles of listings and output thrown away over the years. The computations whose results appear in the tables have generally been at the edge of that which was feasibly computable at the time; between the clever algorithms and the list of factored integers lies an ocean of code and many hours of hard work by people and machines.
This book is a magnificent short compilation of techniques, results, and the modern history of factoring. The Cunningham project has been the focal point for research in factoring, and the book very nicely draws together the threads of the 117 papers listed in the bibliography. (MathSciNet)