Factorizations of b n ±1, Up to High Powers. Third Edition. John Brillhart, D. H. Lehmer J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff, Jr.

Transcription

1 CONTEMPORARY MATHEMATICS 22 Factorizations of b n ±1, b = 2, 3, 5, 6, 7,10, 11, 12 Up to High Powers Third Edition John Brillhart, D. H. Lehmer J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff, Jr. American Mathematical Society Providence, Rhode Island

2 This work is dedicated to the memory of Lt.-Col. Allan J. C. Cunningham ( ), an industrious computer and maker of mathematical tables. This edition is also dedicated to the memory of Derrick Henry Lehmer ( ), our friend, co-author and long-time collaborator.

3 Table of Contents I. Introduction to the Short Tables xi II. Convenient Short Tables Short 2 : 2 n 1, n xiii Short 2+ : 2 n +1, n xxiv Short 10 : 10 n 1, n xxxv Short 10+ : 10 n +1, n xl III. Introduction to the Main Tables A. The Cunningham-Woodall Tables and Their Influence The Cunningham Project xlv B. Developments Contributing to the Present Tables 1. Developments in Technology xlvii 2. Developments in Factorization lv 3. Developments in Primality Testing (a) The Theory lix (b) The Programs lxv (c) The Proof Summaries lxvi C. Multiplicative Structure of b n ± 1 1. Algebraic and Primitive Factors lxviii 2. Aurifeuillian Factorizations lxix D. Acknowledgements lxxiii E. References lxxiv IV. Update to the Introduction for the Second Edition lxxix A. Developments Contributing to the Second Edition 1. Developments in Technology lxxx 2. Developments in Factorization lxxxi 3. Developments in Primality Testing lxxxiv B. Acknowledgements for the Second Edition lxxxv C. References for the Second Edition lxxxvi

4 CONTENTS x V. Update to the Introduction for the Third Edition lxxxix A. Developments Contributing to the Third Edition 1. Developments in Technology lxxxix 2. Developments in Factorization xc 3. Developments in Primality Testing xci B. Status of the Project and of Important Factorizations xcii C. Acknowledgements for the Third Edition xcvi D. References for the Third Edition xcviii VI. How to Use the Main Tables c VII. The Main Tables 2 n 1 n odd n< n +1 n odd n< n +1 n =4k 2 n<2400 L, M n +1 n =4k n n 1 n odd n< n +1 n 540 L, M for n =6k n 1 n odd n<375 L, M for n =10k n +1 n n 1 n odd n< n +1 n 330 L, M for n =12k n 1 n odd n< n +1 n 300 L, M for n =14k n 1 n odd n< n +1 n 330 L, M for n =20k n 1 n odd n< n +1 n 240 L, M for n =22k n 1 n odd n< n +1 n 240 L, M for n =6k VIII. Introduction to the Appendices A. Primes and Probable Primes B. Primality Proof Summaries C. Composite Cofactors

5 I. Introduction to the Short Tables. The following four tables, which contain the known prime factors of the commonly encountered numbers 2 n ± 1and10 n ± 1, have been placed at the beginning of this book for easy reference. Prime factors are given explicitly in these tables and are thus immediately and conveniently at hand. The factorizations are presented differently here than in the main tables, where the factors of a number must usually be collected from various lines in the same or related tables. For example, in the main tables, the 17 prime factors of =(2 15 1)( )( )( ) must be collected from 17 different lines in the four separate base two tables; the 11 prime factors of can all be found in Table 10+, but on 6 different lines. The short tables, of course, may also serve as a check on the reader s ability to use the main tables properly. The notation Pxx in a line represents a prime factor with xx decimal digits, which are given explicitly in Appendix A. For example, the factorization of isgivenas P74 in the Short 2 table. The 74 digits of the large prime factor may be found in Appendix A in the two lines which begin 74 2,269. They appear as When there are more factors than will fit on one line, the extra factors are given on the next line (followed by the line number), the factorization being broken at a multiplication dot, which is repeated on the second line. In Short 2+, for example, the final two prime factors for line 210 are and Note that we use a period rather than a centered dot for multiplication. xi

6 Short 2 Factorizations of 2 n 1, n 400 n Prime Factors xiii

7 Short 2 Prime Factors xiv

8 xv Prime Factors Short

9 Short 2 Prime Factors xvi

10 xvii Prime Factors Short

11 Short 2 Prime Factors xviii

12 xix Prime Factors Short P P

13 Short 2 Prime Factors xx

14 xxi Prime Factors Short P

15 Short 2 Prime Factors xxii P P

16 xxiii Prime Factors Short P P

17 Short 2+ Factorizations of 2 n +1,n 400 n Prime Factors xxiv

18 xxv Prime Factors Short

19 Short 2+ Prime Factors xxvi

20 xxvii Prime Factors Short

21 Short 2+ Prime Factors xxviii

22 xxix Prime Factors Short