The Book of Prime Number Records

495 Pages·1989·15.222 MB·English

by  Paulo Ribenboim (auth.)

The Book of Prime Number Records
Second Edition
Paulo Ribenboim
The Book of
Prime Number Records
Second Edition
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Paulo Ribenboim
Department of Mathematics and Statistics
Queen's University
Kingston, Ontario K7L 3N6
Canada
Mathematics Subject Classification (1980): 10H15, 10H20
Library of Congress Cataloging-in-Publication Data
Ribenboim, Paulo.
The book of prime number records / Paulo Ribenboim--2nd ed.
p. cm.
Includes bibliographical references.
ISBN-13: 978-1-4684-0509-5
1. Numbers, Prime I. Title.
QA246.R47 1989 512'.72--dc20 89-21675
Copyright 1989 Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1989
This work may not be copied in whole or in part without the written permission of the
publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA), except for brief
excerpts in connection with reviews or scholarly analysis. Use in connection with any form of
information storage or retrevial, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc. in this publication, even if
former are not especially identified, is not to be taken as a sign that such names, as understood
by the Trade Marks and Merchandise Act, may accordingly be used freely by anyone.
987654321
ISBN-13: 978-1-4684-0509-5 e-ISBN-13: 978-1-4684-0507-1
DOl: 10.1007/978-1-4684-0507-1
Narrow road to A Jean Pierre Serre, qui
a far province. montre Ie chemin.
BashO
PREFACE TO THE FIRST EDITION
This text originated as a lecture delivered November 20, 1984, at
Queen's University, in the undergraduate colloquim series established
to honor Professors A. J. Coleman and H. W. Ellis and to acknow
ledge their long lasting interest in the quality of teaching under
graduate students.
In another colloquim lecture, my colleague Morris Orzech, who
had consulted the latest edition of the Guilllless Book oj Records,
remainded me very gently that the most "innumerate" people of the
world are of a certain tribe in Mato Grosso, Brazil. They do not
even have a word to express the number "two" or the concept of
plurality. "Yes Morris, I'm from Brazil, but my book will contain
numbers different from 'one.' "
He added that the most boring 800-page book is by two Japanese
mathematicians (whom I'll not name), and consists of about 16
million digits of the number 11.
"I assure you Morris, that in spite of the beauty of the apparent
randomness of the decimal digits of 11, I'll be sure that my text will
include also some words."
Acknowledgment. The manuscript of this book was prepared on the
word processor by Linda Nuttall. I wish to express my appreciation
for the great care, speed, and competence of her work.
Paulo Ribenboim
PREFACE TO THE SECOND EDITION
What is new in this edition?
A few misprints were corrected, as well as a wrong statement
about Mersenne numbers which, embarrassingly, I attributed to Bob
Silverman. There are also several new records in the text, which
were attained since this book appeared one year ago, including a
new Mersenne number. On the other hand, no space could be made
to describe the very recent primality testing methods involving
elliptic curves.
Finally, a probable new record:
RECORD
The fastest selling book on prime number records is ..
You may begin reading it!
Paulo Ribenboim
TABLE OF CONTENTS
Preface to the First Edition vii
Preface to the Second Edition viii
Guiding the Reader xiii
Index of Notations xv
Introduction
Chapter 1. How Many Prime Numbers Are There? 3
I. Euclid's Proof 3
II. Kummer's Proof 4
III. Polya's Proof 5
IV. Euler's Proof 7
V. Thue's Proof 8
VI. Two-and-a-Half Forgotten Proofs 8
A. Perott's Proof 9
B. Auric's Proof 9
C. Metrod's Proof 10
VII. Washington's Proof 10
VIII. FUrstenberg's Proof 11
Chapter 2. How to Recognize Whether a Natural
Number Is a Prime? 13
I. The Sieve of Eratosthenes 14
II. Some Fundamental Theorems on Congruences 15
A. Fermat's Little Theorem and Primitive Roots
Modulo a Prime 16
B. The Theorem of Wilson 19
x Contents
C. The Properties of Giuga, Wolstenholme and Mann
& Shanks 20
D. The Power of a Prime Dividing a Factorial 22
E. The Chinese Remainder Theorem 24
F. Euler's Function 25
G. Sequences of Binomials 31
H. Quadratic Residues 34
III. Classical Primality Tests Based on Congruences 36
IV. Lucas Sequences 41
Addendum on Lehmer Numbers 61
V. Classical Primality Tests Based on Lucas Sequences 61
VI. Fermat Numbers 71
VII. Mersenne Numbers 75
Addendum on Perfect Numbers 81
VIII. Pseudoprimes 86
A. Pseudoprimes in Base 2 (psp) 86
B. Pseudoprimes in Base a (psp(a» 89
B. Euler Pseudoprimes in Base a (epsp(a» 92
D. Strong Pseudoprimes in Base a (spsp(a» 94
Addendum on the Congruence an-k == bn-k (mod n) 97
IX. Carmichael Numbers 98
Addendum on Knodel Numbers 100
X. Lucas Pseudo primes 101
A. Fibonacci Pseudoprimes 102
Q»
B. Lucas Pseudoprimes (Jlpsp(P, 104
Q»
C. Euler-Lucas Pseudoprimes (eJlpsp(P, and Strong
Q»
Lucas Pseudo primes (sJlpsp(P, 105
D. Carmichael-Lucas Numbers 107
XI. Last Section on Primality Testing and Factorization! 107
A. The Cost of Testing 109
B. Recent Primality Tests 113
C. Monte Carlo Methods 116
D. Titanic Primes 120
E. Factorization 121
F. Public Key Cryptography 123
Chapter 3. Are There Functions Defining Prime Numbers? 129
I. Functions Satisfying Condition (a) 129
II. Functions Satisfying Condition (b) 135
III. Functions Satisfying Condition (c) 144
Chapter 4. How Are the Prime Numbers Distributed? 153
I. The Growth of 7l(x) 154
A. History Unfolding 155
Euler 155
Legendre 159
Gauss 160
Contents xi
Tschebycheff 160
Riemann 161
de 1a Vallee Poussin and Hadamard 164
Erdos and Selberg 167
B. Sums Involving the Mobius Function 168
C. The Distribution of Values of Euler's Function 172
D. Tables of Primes 175
E. The Exact Value of n(x) and Comparison with
x/(log x), Li(x), and R(x) 178
F. The Nontrivial Zeroes of t(s) 180
G. Zero-Free Regions for t(s) and the Error Term
in the Prime Number Theorem 184
H. The Growth of t(s) 186
II. The nth Prime and Gaps 188
A. Some Properties of n(x) 188
B. The nth Prime 190
C. Gaps between Primes 191
D. The Possible Gaps between Primes 196
E. Interlude 197
III. Twin Primes 199
Addendum on Polignac's Conjecture 204
IV. Primes in Arithmetic Progression 205
A. There Are Infinitely Many! 205
B. The Smallest Prime in an Arithmetic Progression 217
C. Strings of Primes in Arithmetic Progression 223
V. Primes in Special Sequences 226
VI. Goldbach's Famous Conjecture 229
VII. The Waring-Goldbach Problem 236
A. Waring's Problem 236
B. The Waring-Goldbach Problem 247
VIII. The Distribution of Pseudoprimes and of
Carmichael Numbers 248
A. Distribution of Pseudoprimes 248
B. Distribution of Carmichael Numbers 252
C. Distribution of Lucas Pseudoprimes 253
Chapter 5. Which Special Kinds of Primes Have Been
Considered? 255
I. Regular Primes 255
II. Sophie Germain Primes 261
III. Wieferich Primes 263
IV. Wilson Primes 277
V. Repunits and Similar Numbers 277
VI. Primes with Given Initial and Final Digits 280
VII. Numbers k x 2n ± 1 280
Addendum on Cullen Numbers 283
xii Table of Contents
VIII. Primes and Second-Order Linear Recurrence
Sequences 283
IX. The NSW-Primes 288
Chapter 6. Heuristic and Probabilistic Results About
Prime Numbers 291
I. Prime Values of Linear Polynomials 292
II. Prime Values of Polynomials of Arbitrary Degree 307
III. Some Probabilistic Estimates 321
A. Partitio Numerorum 321
B. Polynomials with Many Successive
Composite Values 329
C. Distribution of Mersenne Primes 332
D. The log log Philosophy 333
IV. The Density of the Set of Regular Primes 334
Conclusion 347
Dear Reader 347
Citations for Some Possible Prizes for Work on
the Prime Number Theorem 349
Bibliography 355
A. General References 355
B. Specific References 357
Chapter 1 357
Chapter 2 361
Chapter 3 390
Chapter 4 398
Chapter 5 439
Chapter 6 449
Conclusion 455
Primes up to 10,000 456
Index of Names 461
Gallima ufr ies 471
Subject Index 473
Addenda to the Second Edition 477