the little book of bigger primes - ribenboim

The Little Book of Bigger Primes
Second Edition
Paulo Ribenboim
The Little Book of
Bigger Primes
Second Edition
Paulo Ribenboim
Department of Mathematics and Statistics
Queen’s University
Kingston, ON K7L 3N6
Canada
Mathematics Subject Classification (2000): 11A41, 11B39, 11A51
Library of Congress Cataloging-in-Publication Data
Ribenboim, Paulo.
The little book of bigger primes / Paulo Ribenboim.
p. cm.
Includes bibliographical references and index.
ISBN 0-387-20169-6 (alk. paper)
1. Numbers, Prime. I. Title.
QA246.R473 2004
512.7′23—dc22 2003066220
ISBN 0-387-20169-6 Printed on acid-free paper.
See first edition  1991 Paulo Ribenboim.
 2004 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag New York, Inc., 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 and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if
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Printed in the United States of America. (EB)
987654321 SPIN 10940969
Springer-Verlag is a part of Springer Science+Business Media
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Nel mezzo del cammin di nostra vita
mi ritrovai per una selva oscura
che la diritta via era smarrita
Dante Alighieri, L’Inferno
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Preface
This book could have been called “Selections from the New Book of
Prime Number Records.” However, I preferred the title which pro-
pelled you on the first place to open it, and perhaps (so I hope) to
buy it!
But the book is not very different from its parent. Like a bonsai,
which has all the main characteristics of the full-sized tree, this pa-
perback should exert the same fatal attraction. I wish it to be as
dangerous as the other one. I wish you, young student, teacher or
retired mathematician, engineer, computer buff, all of you who are
friends of numbers, to be driven into thinking about the beautiful
theory of prime numbers, with its inherent mystery. I wish you to
exercise your brain and fingers—not vice-versa.
This second edition is still a little book, but the primes have
“grown bigger”. An irrepressible activity of computation special-
ists has pushed records to levels previously unthinkable. These en-
deavours generated—or were possible by—new algorithms and great
advances in programming techniques and hardware developments.

A fruitful interplay for the intended aim, to produce large, awesome
numbers.
These updated records are reported; they are like a snapshot taken
May 2003. However, only limited progress was made in the theoret-
ical results. They are explained in the appropriate place. The old
viii Preface
classical problems remain open and continue defying our great minds.
With an inner smile: “If you solve me, you’ll become idle”. Not know-
ing that we, mathematicians, invent more problems than we can
solve. Idle, we shall not be.
Paulo Ribenboim
Acknowledgements
First and foremost, I wish to express my gratitude to Wilfrid Keller.
He spent uncountable hours working on this book, informing me of
the newest records, discussing my text to great depths, with judicious
comments. He also took up the arduous task of preparing the camera-
ready copy. Like the proud Buenos Aires tailor who was not happy
until the jacket fitted to perfection.
I have also obtained great support from many colleagues who ex-
plained patiently their results. As a consequence, their names are
included in the text.
Finally, Chris Caldwell maintains a rich, well selected, informative
website on prime numbers, which I consulted often with great profit.
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Contents
Preface vii
Acknowledgements ix
Guiding the Reader xv
Index of Notations xvii
Introduction 1

1 How Many Prime Numbers Are There? 3
I Euclid’s Proof 3
II Goldbach Did It Too! 6
III Euler’s Proof 8
IV Thue’s Proof 9
V Three Forgotten Proofs 10
A Perott’s Proof 10
B Auric’s Proof 11
CM´etrod’s Proof 11
VI Washington’s Proof 11
VII Furstenberg’s Proof 12
2 How to Recognize Whether a Natural Number is a
Prime 15
I The Sieve of Eratosthenes 16
xii Contents
II Some Fundamental Theorems on Congruences 17
A Fermat’s Little Theorem and Primitive Roots
Modulo a Prime 17
B The Theorem of Wilson 21
C The Properties of Giuga and of Wolstenholme . 21
D The Power of a Prime Dividing a Factorial . . 24
E The Chinese Remainder Theorem 26
F Euler’s Function 28
G Sequences of Binomials 33
H Quadratic Residues 37
III Classical Primality Tests Based on Congruences 39
IV Lucas Sequences 44
V Primality Tests Based on Lucas Sequences 63
VI Fermat Numbers 70
VII Mersenne Numbers 75

VIII Pseudoprimes 88
A Pseudoprimes in Base 2 (psp) 88
B Pseudoprimes in Base a (psp(a)) 92
C Euler Pseudoprimes in Base a (epsp(a)) 95
D Strong Pseudoprimes in Base a (spsp(a)) . . . 96
IX Carmichael Numbers 100
X Lucas Pseudoprimes 103
A Fibonacci Pseudoprimes 104
B Lucas Pseudoprimes (lpsp(P, Q)) 106
C Euler-Lucas Pseudoprimes (elpsp(P, Q)) and
Strong Lucas Pseudoprimes (slpsp(P, Q)) . . . 106
D Carmichael–Lucas Numbers 108
XI Primality Testing and Factorization 109
A The Cost of Testing 110
B More Primality Tests 111
C Titanic and Curious Primes 119
D Factorization 122
E Public Key Cryptography 126
3 Are There Functions Defining Prime Numbers? 131
I Functions Satisfying Condition (a) 131
II Functions Satisfying Condition (b) 137
III Prime-Producing Polynomials 138
A Prime Values of Linear Polynomials 139
Contents xiii
B On Quadratic Fields 140
C Prime-Producing Quadratic Polynomials 144
D The Prime Values and Prime Factors Races . . 148
IV Functions Satisfying Condition (c) 151
4 How Are the Prime Numbers Distributed? 157
I The Function π(x) 158

A History Unfolding 159
B Sums Involving the M¨obius Function 172
C Tables of Primes 173
D The Exact Value of π(x) and Comparison with
x/ log x, Li(x), and R(x) 174
E The Nontrivial Zeros of ζ(s) 177
F Zero-Free Regions for ζ(s) and the Error Term
in the Prime Number Theorem 180
G Some Properties of π(x) 181
H The Distribution of Values of Euler’s Function 183
II The nth Prime and Gaps Between Primes 184
A The nth Prime 185
B Gaps Between Primes 186
III Twin Primes 192
IV Prime k-Tuplets 197
V Primes in Arithmetic Progression 204
A There Are Infinitely Many! 204
B The Smallest Prime in an Arithmetic
Progression 207
C Strings of Primes in Arithmetic Progression . . 209
VI Goldbach’s Famous Conjecture 211
VII The Distribution of Pseudoprimes and of Carmichael
Numbers 216
A Distribution of Pseudoprimes 216
B Distribution of Carmichael Numbers 218
C Distribution of Lucas Pseudoprimes 220
5 Which Special Kinds of Primes Have Been
Considered? 223
I Regular Primes 223
II Sophie Germain Primes 227

III Wieferich Primes 230
IV Wilson Primes 234
xiv Contents
V Repunits 235
VI Numbers k × b
n
± 1 237
VII Primes and Second-Order Linear Recurrence
Sequences 243
6 Heuristic and Probabilistic Results About Prime
Numbers 249
I Prime Values of Linear Polynomials 250
II Prime Values of Polynomials of Arbitrary Degree . . . 253
III Polynomials with Many Successive Composite Values . 261
IV Partitio Numerorum 263
Appendix 1 269
Appendix 2 275
Conclusion 279
Bibliography 281
Web Site Sources 325
Primes up to 10,000 327
Index of Tables 331
Index of Records 333
Index of Names 335
Subject Index 349
Guiding the Reader
If a notation, which is not self-explanatory, appears without explana-
tion on, say, page 107, look at the Index of Notations, which is orga-
nized by page number; the definition of the notation should appear
before or at page 107.

If you wish to see where and how often your name is quoted in this
book, turn to the Index of Names, at the end of the book. Should I
say that there is no direct relation between achievement and number
of quotes earned?
If, finally, you do not want to read the book but you just want
to have some information about Cullen numbers—which is perfectly
legitimate, if not laudable—go quickly to the Subject Index. Do not
look under the heading Numbers, but rather Cullen. And for a sub-
ject like Strong Lucas pseudoprimes, you have exactly three possibil-
ities .
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Index of Notations
The following traditional notations are used in the text without ex-
planation:
Notation Explanation
m | n the integer m divides the integer n
m  n the integer m does not divide the integer n
p
e
 npis a prime, p
e
| n but p
e+1
 n
gcd(m, n) greatest common divisor of the integers m, n
lcm(m, n) least common multiple of the integers m, n
log x natural logarithm of the real number x>0
Z ring of integers
Q field of rational numbers
R field of real numbers

C field of complex numbers
The following notations are listed as they appear in the book:
xviii Index of Notations
Page Notation Explanation
3 p
n
the nth prime
4 p# product of all primes q ≤ p,or
primorial of p
6 F
n
nth Fermat number, F
n
=2
2
n
+1
15 [x] the largest integer in x, that is, the only
integer [x] such that [x] ≤ x<[x]+1
19 g
p
smallest primitive root modulo p
28 ϕ(n) totient or Euler’s function
29 λ(n) Carmichael’s function
30 ω(n) number of distinct prime factors of n
30 L(x) number of composite n such that n ≤ x
and ϕ(n) divides n − 1
31 V
ϕ
(m)#{n ≥ 1 | ϕ(n)=m}

34 t
∗
n
primitive part of a
n
− b
n
35 k(m) square-free kernel of m
36 P [m] largest prime factor of m
36 S
r
set of integers n with at most r log log n
distinct prime factors
37

a
p

or (a | p) Legendre symbol
38

a
b

or (a | b) Jacobi symbol
44 U
n
= U
n
(P, Q) nth term of the Lucas sequence with

parameters (P, Q)
44 V
n
= V
n
(P, Q) nth term of the companion Lucas
sequence with parameters (P, Q)
50 ρ(n)=ρ(n, U ) smallest r ≥ 1 such that n divides U
r
50 ψ(p)=p − (D | p)
52

α, β
p

a symbol associated to the roots α, β
of X
2
− PX + Q
52 ψ
α,β
(p)=p −

α, β
p

, where p is an odd prime
52 ψ
α,β
(p

e
)=p
e−1
ψ
α,β
(p), where e ≥ 1 and p is
an odd prime
52 λ
α,β


p
e

lcm{ψ
α,β
(p
e
)}
56 P(U) set of primes p dividing some term U
n
=0
56 P(V ) set of primes p dividing some term V
n
=0
Index of Notations xix
Page Notation Explanation
58 U
∗
n

primitive part of U
n
63 ψ
D

s

i=1
p
e
i
i

=
1
2
s−1
s

i=1
p
e
i
−1
i

p
i
−


D
p
i

72 Pn prime number with n digits
72 Cn composite number with n digits
76 M
q
=2
q
− 1, Mersenne number
84 σ(n) sum of divisors of n
86 τ(n) number of divisors of n
86 H(n) harmonic mean of divisors of n
86 V (x)#{N perfect number | N ≤ x}
87 s(n) sum of aliquot parts of n
88 psp pseudoprime in base 2
92 psp(a) pseudoprime in base a
94 B
psp
(n) number of bases a,1<a≤ n −1,
gcd(a, n) = 1, such that n is a psp(a)
95 epsp(a) Euler pseudoprime in base a
96 B
epsp
(n) number of bases a,1<a≤ n − 1,
gcd(a, n) = 1, such that n is a epsp(a)
97 spsp(a) strong pseudoprime in base a
98 B
spsp

(n) number of bases a,1<a≤ n − 1,
gcd(a, n) = 1, such that n is a spsp(a)
101 M
3
(m)=(6m + 1)(12m + 1)(18m +1)
101 M
k
(m)=(6m + 1)(12m +1)
k−2

i=1
(9 × 2
i
m +1)
102 C
k
set of all composite integers n>k
such that if 1 <a<n, gcd(a, n)=1,
then a
n−k
≡ 1 (mod n)
(the Kn¨odel numbers when k>1)
103 lpsp(P, Q) Lucas pseudoprime with parameters (P, Q)
106 B
lpsp
(n, D) number of integers P ,1≤ P ≤ n,
such that there exists Q with P
2
− 4Q
≡ D (mod n) and n is a lpsp(P, Q)

106 elpsp(P, Q) Euler-Lucas pseudoprime with parameters
(P, Q)
107 slpsp(P, Q) strong Lucas pseudoprime with
parameters (P, Q)
132 π(x) the number of primes p ≤ x
xx Index of Notations
Page Notation Explanation
134 µ(x)M¨obius function
140 ∆ fundamental discriminant associated
to d =0, 1
140 Q(
√
d)=Q(
√
∆), quadratic field
141 Cl
d
or Cl
∆
class group of Q(
√
d)
141 h
d
or h
∆
class number of Q(
√
d)
141 e

d
exponent of the class group Cl
d
148 π
∗
f(X)
(N)#{n | 0 ≤ n ≤ N,|f(n)| is a prime}
150 P
0
[m] smallest prime factor of m>1
150 P
0
[f(X)] = min{P
0
[f(k)] | k =0, 1, 2, }
158 f(x) ∼ h(x) f, h are asymptotically equal
159 f(x)=g(x) the difference f(x) −g(x) is ultimately
+O(h(x)) bounded by a constant multiple of h(x)
159 f(x)=g(x) the difference f(x) −g(x) is negligible
+o(h(x)) in comparison to h(x)
159 ζ(s) Riemann’s zeta function
161 B
k
Bernoulli number
162 S
k
(n)=

n
j=1

j
k
162 B
k
(X) Bernoulli polynomial
163 Li(x) logarithmic integral
164 θ(x)=

p≤x
log p, Tschebycheff’s function
165 Re(s) real part of s
165 Γ(s) gamma function
166 γ Euler’s constant
167 J(x) weighted prime-power counting function
168 R(x) Riemann’s function
169 Λ(x) von Mangoldt’s function
170 ψ(x) summatory function of the von Mangoldt
function
172 M(x) Mertens’ function
174 ϕ(x, m)#{a | 1 ≤ a ≤ x, a is not a multiple of
2, 3, ,p
m
}
177 ρ
n
nth zero of ζ(s) in the upper half of the
critical strip
177 N(T )#{ρ = σ + it | 0 ≤ σ ≤ 1,ζ(ρ)=0,
0 <t≤ T }
186 d

n
= p
n+1
− p
n
Index of Notations xxi
Page Notation Explanation
186 g(p) number of successive composite integers
greater than p
186 G = {m | m = g(p) for some p>2}
186 p[m] the smallest prime p such that g(p)=m
191 log
2
x log log x
191 log
3
x log log log x
191 log
4
x log log log log x
193 B Brun’s constant
194 π
2
(x)#{p prime | p +2≤ x and p + 2 is also
a prime}
194 C
2
=

p>2


1 −
1
(p − 1)
2

, twin prime constant
197 π
2k
(x)#{n ≥ 1 | p
n
≤ x and p
n+1
− p
n
=2k}
198 π
2,6
(x)#{p prime | p ≤ x and p +2,p+ 6 are also
primes}
198 π
4,6
(x)#{p prime | p ≤ x and p +4,p+ 6 are also
primes}
198 π
2,6,8
(x)#{p prime | p ≤ x and p +2,p+6,p+8
are also primes}
198 B
2,6

=


1
p
+
1
p +2
+
1
p +6

,
the summation extended over all triplets
of primes (p, p +2,p+6)
198 B
4,6
=


1
p
+
1
p +4
+
1
p +6

,

the summation extended over all triplets
of primes (p, p +4,p+6)
198 B
2,6,8
=


1
p
+
1
p +2
+
1
p +6
+
1
p +8

,
summation over all quadruplets of
primes (p, p +2,p+6,p+8)
200 ρ
∗
(x)=k if there exists an admissible
(k −1)-tuple below x, but none
with more components
200 ρ(x) = lim sup
y→∞


π(x + y) −π(y)

205 π
d,a
(x)#{p prime | p ≤ x, p ≡ a (mod d)}
207 p(d, a) smallest prime in the arithmetic
progression {a + kd | k ≥ 0}
xxii Index of Notations
Page Notation Explanation
207 p(d) = max{p(d, a) | 1 ≤ a<d,gcd(a, d)=1}
207 L Linnik’s constant
212 P
k
set of all k-almost-primes
213 S, S
0
Schnirelmann’s constants
214 r
2
(2n) number of representations of 2n as sum
of two primes
215 G

(n)#{2n | 2n ≤ x, 2n is not a sum of two
primes
216 (psp)
n
nth pseudoprime
216 Pπ(x) number of pseudoprimes to base 2,
less than or equal to x

217 Pπ
a
(x) same, to base a
217 EPπ(x) number of Euler pseudoprimes to base 2,
less than or equal to x
217 EPπ
a
(x) same, to base a
217 SPπ(x) number of strong pseudoprimes to base 2,
less than or equal to x
217 SPπ
a
(x) same, to base a
217 l(x)=e
log x log log log x/ log log x
218 psp(d, a) smallest pseudoprime in the arithmetic
progression {a + kd | k ≥ 1} with
gcd(a, d)=1
218 CN(x)#{n | 1 ≤ n ≤ x, n Carmichael number}
220 Lπ(x) number of Lucas pseudoprimes with
parameters (P, Q), less than or equal x
221 SLπ(x) number of strong Lucas pseudoprimes with
parameters (P, Q), less than or equal x
224 ζ
p
= cos(2π/p)+i sin(2π/p)
225 h(p) class number of the pth cyclotomic field
226 π
reg
(x) number of regular primes p ≤ x

226 π
ir
(x) number of irregular primes p ≤ x
226 ii(p) irregularity index of p
226 π
iis
(x) number of primes p ≤ x such that ii(p)=s
228 S
d,a
(x)#{p prime | p ≤ x, dp + a is a prime}
231 q
p
(a)=
a
p−1
− 1
p
, Fermat quotient of p with base a
234 W (p)=
(p − 1)!+1
p
, Wilson quotient
Index of Notations xxiii
Page Notation Explanation
235 Rn =
10
n
− 1
9
, repunit

240 Cn = n × 2
n
+ 1, Cullen number
241 Cπ(x)#{n | Cn ≤ x and Cn is prime}
241 Wn = n × 2
n
− 1, Woodall number or Cullen
number of the second kind
244 P(T ) set of primes p dividing some term of the
sequence T =(T
n
)
n≥0
244 π
H
(x)#{p ∈P(H) | p ≤ x}
247 S
2n+1
NSW number
254 π
f(X)
(x)#{n ≥ 1 ||f(n)|≤x and |f(n)| is prime}
261 p(f) smallest integer m ≥ 1 such that |f(m)|
is a prime
263 π
X,X+2k
(x)#{p prime | p +2k prime and p +2k ≤ x}
264 π
X
2

+1
(x)#{p prime | p is of the form p = m
2
+1
and p ≤ x}
265 π
aX
2
+bX+c
(x)#{p prime | p is of the form
p = am
2
+ bm + c and p ≤ x}
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