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Title
Dedication
Table of Contents
I. Introduction to the Short Tables
II. Convenient Short Tables
III. Introduction to the Main Tables
IV. Update to the Introduction for the Second Edition
V. Update to the Introduction for the Third Edition
VI. How to Use the Main Tables
VII. The Main Tables
VIII. Introduction to the Appendices
Nội dung
American Mathematical Society
Providence, Rhode Island
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.
22
C
ONTEMPORARY
M
ATHEMATICS
This work is dedicated to the memory of
Lt Col. Allan J. C. Cunningham
(1842-1928), an industrious computer
and maker of mathematical tables.
This edition is also dedicated to the memory of
Derrick Henry Lehmer (1905-1991),
our friend, co-author and long-time collaborator.
Table of Contents
I.IntroductiontotheShortTables xi
II. Convenient Short Tables
Short 2− :2
n
− 1,n≤ 400 xiii
Short 2+ : 2
n
+1,n≤ 400 xxiv
Short 10− :10
n
− 1,n≤ 150 xxxv
Short 10+ : 10
n
+1,n≤ 150 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.DevelopmentsinTechnology xlvii
2.DevelopmentsinFactorization lv
3. Developments in Primality Testing
(a)TheTheory lix
(b)ThePrograms lxv
(c)TheProofSummaries lxvi
C. Multiplicative Structure of b
n
± 1
1.AlgebraicandPrimitiveFactors lxviii
2. Aurifeuillian Factorizations . lxix
D.Acknowledgements lxxiii
E.References lxxiv
IV.UpdatetotheIntroductionfortheSecondEdition lxxix
A. Developments Contributing to the Second Edition
1.DevelopmentsinTechnology lxxx
2.DevelopmentsinFactorization lxxxi
3.DevelopmentsinPrimalityTesting lxxxiv
B.AcknowledgementsfortheSecondEdition lxxxv
C.ReferencesfortheSecondEdition lxxxvi
CONTENTS x
V.UpdatetotheIntroductionfortheThirdEdition lxxxix
A. Developments Contributing to the Third Edition
1.DevelopmentsinTechnology lxxxix
2.DevelopmentsinFactorization xc
3.DevelopmentsinPrimalityTesting xci
B.StatusoftheProjectandofImportantFactorizations xcii
C.AcknowledgementsfortheThirdEdition xcvi
D.ReferencesfortheThirdEdition xcviii
VI.HowtoUsetheMainTables c
VII. The Main Tables
2
n
− 1 n odd n<1200 . . 1
2
n
+1 n odd n<1200 . . 14
2
n
+1 n =4k −2 n<2400 L, M 27
2
n
+1 n =4kn≤ 1200 . . 53
3
n
− 1 n odd n<540 60
3
n
+1 n ≤ 540 L, M for n =6k − 3 ≤ 1077 66
5
n
− 1 n odd n<375 L, M for n =10k − 5 ≤ 745 86
5
n
+1 n ≤ 375 94
6
n
− 1 n odd n<330 102
6
n
+1 n ≤ 330 L, M for n =12k − 6 ≤ 654 106
7
n
− 1 n odd n<300 116
7
n
+1 n ≤ 300 L, M for n =14k − 7 ≤ 595 120
10
n
− 1 n odd n<330 129
10
n
+1 n ≤ 330 L, M for n =20k − 10 ≤ 650 133
11
n
− 1 n odd n<240 142
11
n
+1 n ≤ 240 L, M for n =22k − 11 ≤ 473 145
12
n
− 1 n odd n<240 152
12
n
+1 n ≤ 240 L, M for n =6k − 3 ≤ 477 155
VIII.IntroductiontotheAppendices 165
A.PrimesandProbablePrimes 167
B.PrimalityProofSummaries 189
C.CompositeCofactors 235
I. Introduction to the Short Tables.
The following four tables, which contain the known prime factors of the com-
monly 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
120
−1=(2
15
− 1)(2
15
+ 1)(2
30
+ 1)(2
60
+1)
must be collected from 17 different lines in the four separate base two tables; the
11 prime factors of 10
70
+ 1 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 dig-
its, which are given explicitly in Appendix A. For example, the factorization of
2
269
− 1isgivenas“ .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
Short 2−
Factorizations of 2
n
− 1, n ≤ 400
n Prime Factors
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Short 2− Prime Factors xiv
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