my numbers, my friends - popular lectures on number theory

My Numbers, My Friends
Paulo Ribenboim
My Numbers, My Friends
Popular Lectures on Number Theory
Paulo Ribenboim
Department of Mathematics
and Statistics
Queen’s University
Kingston, Ontario K7L 3N6
Canada
Mathematics Subject Classification (2000): 11-06, 11Axx
Library of Congress Cataloging-in-Publication Data
Ribenboim, Paulo
My numbers, my friends / Paulo Ribenboim
p. cm.
Includes bibliographical references and index.
ISBN 0-387-98911-0 (sc. : alk. paper)
1. Number Theory. I. Title
QA241.R467 2000
612’.7— dc21 99-42458
c
 2000 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 Av-
enue, 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
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The use of general descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that such

names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly
be used freely by anyone.
ISBN 0-387-98911-0 Springer-Verlag New York Berlin Heidelberg SPIN 10424971
Contents
Preface xi
1 The Fibonacci Numbers and the Arctic Ocean 1
1 Basicdefinitions . 2
A. Lucassequences . 2
B. SpecialLucassequences . 3
C. Generalizations . 3
2 Basicproperties . 5
A. Binet’sformulas . 5
B. DegenerateLucassequences . 5
C. Growth and numerical calculations . . . . . 6
D. Algebraicrelations . 7
E. Divisibility properties . 9
3 Prime divisors of Lucas sequences . 10
A. The sets P(U), P(V ), and the rank of
appearance . 10
B. Primitive factors of Lucas sequences . . . . 17
4 PrimesinLucassequences . 26
5 Powers and powerful numbers in Lucas sequences . 28
A. Generaltheoremsforpowers . 29
B. Explicit determination in special sequences . 30
vi Contents
C. Uniform explicit determination of
multiples, squares, and square-classes for
certain families of Lucas sequences . . . . . 35
D. Powerful numbers in Lucas sequences . . . . 41
2 Representation of Real Numbers by Means of

Fibonacci Numbers 51
3 Prime Number Records 62
4 Selling Primes 78
5 Euler’s Famous Prime Generating Polynomial 91
1 Quadraticextensions . 94
2 Ringsofintegers . 94
3 Discriminant . 95
4 Decompositionofprimes . 96
A. Propertiesofthenorm . 96
5 Units . 100
6 Theclassnumber . 101
A. Calculationoftheclassnumber . 103
B. Determination of all quadratic fields with
classnumber1 . 106
7 Themaintheorem . 108
6 Gauss and the Class Number Problem 112
1 Introduction . 112
2 HighlightsofGauss’life . 112
3 Briefhistoricalbackground . 114
4 Binaryquadraticforms . 115
5 The fundamental problems . . 118
6 Equivalenceofforms . 118
7 Conditional solution of the fundamental problems . 120
8 Proper equivalence classes of definite forms . . . . . 122
A. Anothernumericalexample . 126
9 Proper equivalence classes of indefinite forms . . . 126
A. Anothernumericalexample . 131
10 Theautomorphofaprimitiveform . 131
11 Composition of proper equivalence classes of
primitiveforms . 135

Contents vii
12 Thetheoryofgenera . 137
13 The group of proper equivalence classes of
primitiveforms . 143
14 Calculationsandconjectures . 145
15 The aftermath of Gauss (or the “math” after
Gauss) . 146
16 Formsversusidealsinquadraticfields . 146
17 Dirichlet’sclassnumberformula . 153
18 Solution of the class number problem for definite
forms . 157
19 The class number problem for indefinite forms . . . 161
20 Morequestionsandconjectures . 164
21 Many topics have not been discussed . 168
7 Consecutive Powers 175
1 Introduction . 175
2 History . 177
3 Specialcases . 178
4 Divisibility properties . 190
5 Estimates . 195
A. The equation a
U
− b
V
=1 . 196
B. The equation X
m
− Y
n
=1 . 197

C. The equation X
U
− Y
V
=1 . 201
6 Final comments and applications . 204
8 1093 213
A. Determination of the residue of q
p
(a) . . . . 216
B. Identities and congruences for the Fermat
quotient . 217
9 Powerless Facing Powers 229
1 Powerfulnumbers . 229
A. Distributionofpowerfulnumbers . 230
B. Additiveproblems . 232
C. Differenceproblems . 233
2 Powers . 235
A. Pythagorean triples and Fermat’s problem . 235
B. VariantsofFermat’sproblem . 238
C. TheconjectureofEuler . 239
D. The equation AX
l
+ BY
m
= CZ
n
. 240
viii Contents
E. Powersasvaluesofpolynomials . 245

3 Exponentialcongruences . 246
A. TheWieferichcongruence . 246
B. Primitivefactors . 248
4 Dreammathematics . 251
A. The statements . 251
B. Statements . 252
C. Binomials and Wieferich congruences . . . . 254
D. Erd¨os conjecture and Wieferich congruence . 257
E. Thedreaminthedream . 257
10 What Kind of Number Is
√
2
√
2
? 271
0 Introduction . 271
1 Kindsofnumbers . 271
2 Hownumbersaregiven . 276
3 Briefhistoricalsurvey . 284
4 Continuedfractions . 287
A. Generalities . 288
B. Periodiccontinuedfractions . 289
C. Simple continued fractions of π and e . . . . 291
5 Approximationbyrationalnumbers . 295
A. Theorderofapproximation . 295
B. TheMarkoffnumbers . 296
C. Measuresofirrationality . 298
D. Order of approximation of irrational
algebraicnumbers . 299
6 Irrationalityofspecialnumbers . 301

7 Transcendental numbers . 309
A. Liouville numbers . . . 310
B. Approximation by rational numbers:
sharpertheorems . 311
C. Hermite, Lindemann, and Weierstrass . . . 316
D. AresultofSiegelonexponentials . 318
E. Hilbert’s7thproblem . 320
F. TheworkofBaker . 321
G. TheconjectureofSchanuel . 323
H. Transcendence measure and the
classificationofMahler . 328
8 Final comments . 331
Contents ix
11 Galimatias Arithmeticae 344
Index of Names 361
Index of Subjects 369
Preface
Dear Friends of Numbers:
This little book is for you. It should offer an exquisite intel-
lectual enjoyment, which only relatively few fortunate people can
experience.
May these essays stimulate your curiosity and lead you to books
and articles where these matters are discussed at a more technical
level.
I warn you, however, that the problems treated, in spite of be-
ing easy to state, are for the most part very difficult. Many are
still unsolved. You will see how mathematicians have attacked these
problems.
Brains at work! But do not blame me for sleepless nights (I have
mine already).

Several of the essays grew out of lectures given over the course of
years on my customary errances.
Other chapters could, but probably never will, become full-sized
books.
The diversity of topics shows the many guises numbers take to
tantalize
∗
and to demand a mobility of spirit from you, my reader,
who is already anxious to leave this preface.
Now go to page 1 (or 127?).
Paulo Ribenboim
∗
Tantalus, of Greek mythology, was punished by continual disappointment
when he tried to eat or drink what was placed within his reach.
1
The Fibonacci Numbers and
the Arctic Ocean
Introduction
There is indeed not much relation between the Fibonacci numbers
and the Arctic Ocean, but I thought that this title would excite your
curiosity for my lecture. You will be disappointed if you wished to
hear about the Arctic Ocean, as my topic will be the sequence of
Fibonacci numbers and similar sequences.
Like the icebergs in the Arctic Ocean, the sequence of Fibonacci
numbers is the most visible part of a theory which goes deep: the
theory of linear recurring sequences.
The so-called Fibonacci numbers appeared in the solution of a
problem by Fibonacci (also known as Leonardo Pisano), in his
book Liber Abaci (1202), concerning reproduction patterns of rab-
bits. The first significant work on the subject is by Lucas, with his

seminal paper of 1878. Subsequently, there appeared the classical
papers of Bang (1886) and Zsigmondy (1892) concerning prime
divisions of special sequences of binomials. Carmichael (1913)
published another fundamental paper where he extended to Lucas se-
quences the results previously obtained in special cases. Since then, I
note the work of Lehmer, the applications of the theory in primality
tests giving rise to many developments.
2 1. The Fibonacci Numbers and the Arctic Ocean
The subject is very rich and I shall consider here only certain
aspects of it.
If, after all, your only interest is restricted to Fibonacci and Lucas
numbers, I advise you to read the booklets by Vorob’ev (1963),
Hoggatt (1969), and Jarden (1958).
1 Basic definitions
A. Lucas sequences
Let P , Q be non-zero integers, let D = P
2
− 4Q, be called the
discriminant, and assume that D = 0 (to exclude a degenerate case).
Consider the polynomial X
2
− PX + Q, called the characteristic
polynomial, which has the roots
α =
P +
√
D
2
and β =
P −

√
D
2
.
Thus, α = β, α + β = P , α ·β = Q, and (α − β)
2
= D.
For each n ≥ 0, define U
n
= U
n
(P, Q)andV
n
= V
n
(P, Q)as
follows:
U
0
=0 ,U
1
=1 ,U
n
=P ·U
n−1
− Q ·U
n−2
(for n ≥ 2),
V
0

=2 ,V
1
=P, V
n
=P ·V
n−1
− Q ·V
n−2
(for n ≥ 2).
The sequences U =(U
n
(P, Q))
n≥0
and V =(V
n
(P, Q))
n≥0
are
called the (first and second) Lucas sequences with parameters (P, Q).
(V
n
(P, Q))
n≥0
is also called the companion Lucas sequence with
parameters (P, Q).
It is easy to verify the following formal power series developments,
for any (P, Q):
X
1 − PX + QX
2

=
∞

n=0
U
n
X
n
and
2 − PX
1 − PX + QX
2
=
∞

n=0
V
n
X
n
.
The Lucas sequences are examples of sequences of numbers
produced by an algorithm.
At the nth step, or at time n, the corresponding numbers are
U
n
(P, Q), respectively, V
n
(P, Q). In this case, the algorithm is a linear
1 Basic definitions 3

recurrence with two parameters. Once the parameters and the initial
values are given, the whole sequence—that is, its future values—is
completely determined. But, also, if the parameters and two consec-
utive values are given, all the past (and future) values are completely
determined.
B. Special Lucas sequences
I shall repeatedly consider special Lucas sequences, which are im-
portant historically and for their own sake. These are the sequences
of Fibonacci numbers, of Lucas numbers, of Pell numbers, and other
sequences of numbers associated to binomials.
(a) Let P =1,Q = −1, so D =5.ThenumbersU
n
= U
n
(1, −1)
are called the Fibonacci numbers , while the numbers V
n
= V
n
(1, −1)
are called the Lucas numbers . Here are the initial terms of these
sequences:
Fibonacci numbers : 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
Lucas numbers : 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 99, 322,
(b) Let P =2,Q = −1, so D =8.ThenumbersU
n
= U
n
(2, −1)
and V

n
= V
n
(2, −1) are the Pell numbers and the companion Pell
numbers. Here are the first few terms of these sequences:
U
n
(2, −1): 0, 1, 2, 5, 12, 29, 70, 169,
V
n
(2, −1): 2, 2, 6, 14, 34, 82, 198, 478,
(c) Let a, b be integers such that a>b≥ 1. Let P = a + b, Q = ab,
so D =(a − b)
2
.Foreachn ≥ 0, let U
n
=
a
n
−b
n
a−b
and V
n
= a
n
+ b
n
.
Then it is easy to verify that U

0
=0,U
1
=1,V
0
=2,V
1
= a+b = P ,
and (U
n
)
n≥0
,(V
n
)
n≥0
are the first and second Lucas sequences with
parameters P , Q.
In particular, if b = 1, one obtains the sequences of numbers U
n
=
a
n
−1
a−1
, V
n
= a
n
+ 1; now the parameters are P = a +1, Q = a.

Finally, if also a = 2, one gets U
n
=2
n
− 1, V
n
=2
n
+ 1, and now
the parameters are P =3,Q =2.
C. Generalizations
At this point, it is appropriate to indicate extensions of the notion
of Lucas sequences which, however, will not be discussed in this
lecture. Such generalizations are possible in four directions, namely,
4 1. The Fibonacci Numbers and the Arctic Ocean
by changing the initial values, by mixing two Lucas sequences, by
not demanding that the numbers in the sequences be integers, or by
having more than two parameters.
Even though many results about Lucas sequences have been ex-
tended successfully to these more general sequences, and have found
interesting applications, for the sake of definiteness I have opted to
restrict my attention only to Lucas sequences.
(a) Let P, Q be integers, as before. Let T
0
, T
1
be any integers such
that T
0
or T

1
is non-zero (to exclude the trivial case). Let
W
0
= PT
0
+2T
1
and W
1
=2QT
0
+ PT
1
.
Let
T
n
= P · T
n−1
− Q ·T
n−2
and
W
n
= P · W
n−1
− Q ·W
n−2
(for n ≥ 2).

The sequences (T
n
(P, Q))
n≥0
and W
n
(P, Q))
n≥0
are the (first and
the second) linear recurrence sequences with parameters (P, Q)and
associated to the pair (T
0
,T
1
). The Lucas sequences are special, nor-
malized, linear recurrence sequences with the given parameters; they
are associated to (0, 1).
(b) Lehmer (1930) considered the following sequences. Let P , Q be
non-zero integers, α, β the roots of the polynomial X
2
−
√
P ·X +Q,
and define
L
n
(P, Q)=










α
n
− β
n
α − β
if n is odd,
α
n
− β
n
α
2
− β
2
if n is even.
L =(L
n
(P, Q))
n≥0
is the Lehmer sequence with parameters P , Q.
Its elements are integers. These sequences have been studied by
Lehmer and subsequently by Schinzel and Stewart in several
papers which also deal with Lucas sequences and are quoted in the
bibliography.

(c) Let R be an integral domain which need not be Z.LetP , Q ∈R,
P , Q = 0, such that D = P
2
−4Q = 0. The sequences (U
n
(P, Q))
n≥0
,
(V
n
(P, Q))
n≥0
of elements of R maybedefinedasforthecasewhen
R = Z.
Noteworthy cases are when R is the ring of integers of a number
field (for example, a quadratic number field), or R = Z[x] (or other
2 Basic properties 5
polynomial ring), or R is a finite field. For this latter situation, see
Selmer (1966).
(d) Let P
0
, P
1
, , P
k−1
(with k ≥ 1) be given integers, usually
subjected to some restrictions to exclude trivial cases. Let S
0
, S
1

,
, S
k−1
be given integers. For n ≥ k, define:
S
n
= P
0
· S
n−1
− P
1
· S
n−2
+ P
2
· S
n−3
− +(−1)
k−1
P
k−1
· S
n−k
.
Then (S
n
)
n≥0
is called a linear recurrence sequence of order k,with

parameters P
0
, P
1
, , P
k−1
and initial values S
0
, S
1
, , S
k−1
.
The case when k = 2 was seen above. For k = 1, one obtains the
geometric progression (S
0
· P
n
0
)
n≥0
.
There is great interest and still much to be done in the theory of
linear recurrence sequences of order greater than 2.
2 Basic properties
The numbers in Lucas sequences satisfy many, many properties that
reflect the regularity in generating these numbers.
A. Binet’s formulas
Binet (1843) indicated the following expression in terms of the roots
α, β of the polynomial X

2
− PX + Q:
(2.1) Binet’s formulas:
U
n
=
α
n
− β
n
α − β
,V
n
= α
n
+ β
n
.
The proof is, of course, very easy. Note that by Binet’s formulas,
U
n
(−P, Q)=(−1)
n−1
U
n
(P, Q)and
V
n
(−P, Q)=(−1)
n

V
n
(P, Q).
So, for many of the following considerations, it will be assumed that
P ≥ 1.
B. Degenerate Lucas sequences
Let (P, Q) be such that the ratio η = α/β of roots of X
2
− Px+ Q
is a root of unity. Then the sequences U(P, Q), V (P, Q)aresaidto
be degenerate.
6 1. The Fibonacci Numbers and the Arctic Ocean
Now I describe all degenerate sequences. Since
η + η
−1
=
α
β
+
β
α
=
P
2
− 2Q
Q
is an algebraic integer and rational, it is an integer. From |
α
β
+

β
α
|≤2
it follows P
2
− 2Q =0,±Q, ±2Q, and this gives P
2
= Q,2Q,3Q,
4Q.Ifgcd(P,Q) = 1, then (P, Q)=(1, 1), (−1, 1), (2, 1), or (−2, 1),
and the sequences are
U(1, 1):0,1,1,0,−1, −1, 0, 1, 1, 0,
U(−1, 1) : 0, 1, −1, 0, 1, −1, 0,
V (1, 1) : 2, 1, −1, −2, −1, 1, 2, 1, −1, −2,
V (−1, 1) : 2, −1, −1, 2, −1, −1, 2,
U(2, 1):0,1,2,3,4,5,6,7,
U(−2, 1) : 0, 1, −2, 3, −4, 5, −6, 7,
V (2, 1):2,2,2,2,2,2,2,2,
V (−2, 1) : 2, −2, 2, −2, 2, −2, 2, −2,
From the discussion, if the sequence is degenerate, then D =0or
D = −3.
C. Growth and numerical calculations
First, I note results about the growth of the sequence U(P, Q).
(2.2) If the sequences U(P,Q), V (P, Q) are non-degenerate, then
|U
n
|, |V
n
| tend to infinity (as n tends to ∞).
This follows from a result of Mahler (1935) on the growth of
coefficients of Taylor series. Mahler also showed

(2.3) If Q ≥ 2, gcd(P, Q)=1,D<0, then, for every ε>0andn
sufficiently large,
|U
n
|≥|β
n
|
1−ε
.
The calculations of U
n
, V
n
may be performed as follows. Let
M =

P −Q
10

.
Then for n ≥ 1,

U
n
U
n−1

= M
n−1


1
0

2 Basic properties 7
and

V
n
V
n−1

= M
n−1

2
P

.
To compute a power M
k
of the matrix M, the quickest method is to
compute successively the powers M, M
2
, M
4
, , M
2
e
where 2
e

≤
k<2
e+1
; this is done by successively squaring the matrices. Next, if
the 2-adic development of k is k = k
0
+k
1
×2+k
2
×2
2
+ +k
e
×2
e
,
where k
i
= 0 or 1, then M
k
= M
k
0
× (M
2
)
k
1
× × (M

2
e
)
k
e
.
Note that the only factors actually appearing are those where k
i
=
1.
Binet’s formulas allow also, in some cases, a quick calculation of U
n
and V
n
.
If D ≥ 5and|β| < 1, then




U
n
−
α
n
√
D





<
1
2
(for n ≥ 1),
and |V
n
− α
n
| <
1
2
(for n such that n · (−log |β|) > log 2). Hence,
cU
n
is the closest integer to
α
n
√
D
,andV
n
is the closest integer to α
n
.
This applies in particular to Fibonacci and Lucas numbers for which
D =5,α =(1+
√
5)/2=1.616 , (the golden number), β =
(1 −

√
5)/2=−0.616
It follows that the Fibonacci number U
n
and the Lucas number V
n
have approximately n/5 digits.
D. Algebraic relations
The numbers in Lucas sequences satisfy many properties. A look at
the issues of The Fibonacci Quarterly will leave the impression that
there is no bound to the imagination of mathematicians whose en-
deavor it is to produce newer forms of these identities and properties.
Thus, there are identities involving only the numbers U
n
, in others
only the numbers V
n
appear, while others combine the numbers U
n
and V
n
. There are formulas for U
m+n
, U
m−n
, V
m+n
, V
m−n
(in terms

of U
m
, U
n
, V
m
, V
n
); these are the addition and subtraction formulas.
There are also formulas for U
kn
, V
kn
,andU
n
k
, V
n
k
, U
k
n
, cV
k
n
(where
k ≥ 1) and many more.
I shall select a small number of formulas that I consider most
useful. Their proofs are almost always simple exercises, either by
applying Binet’s formulas or by induction.

8 1. The Fibonacci Numbers and the Arctic Ocean
It is also convenient to extend the Lucas sequences to negative
indices in such a way that the same recursion (with the given
parameters P, Q) still holds.
(2.4) Extension to negative indices:
U
−n
= −
1
Q
n
U
n
,V
−n
=
1
Q
n
V
n
(for n ≥ 1).
(2.5) U
n
and V
n
maybeexpressedintermsofP , Q. For example,
U
n
= P

n−1
−

n − 2
1

P
n−3
Q +

n − 3
2

P
n−5
Q
2
+
+(−1)
k

n − 1 −k
k

P
n−1−2k
Q
k
+ ···+ (last summand)
where

(last summand) =







(−1)
n
2
−1

n
2
n
2
− 1

PQ
n
2
−1
if n is even,
(−1)
n−1
2
Q
n−1
2

if n is odd.
Thus, U
n
= f
n
(P, Q), where f
n
(X, Y ) ∈ Z[X, Y ]. The function f
n
is
isobaric of weight n −1, where X has weight 1 and Y has weight 2.
Similarly, V
n
= g
n
(P, Q), where g
n
∈ Z[X, Y ]. The function g
n
is
isobaric of weight n,whereX has weight 1, and Y has weight 2.
(2.6) Quadratic relations:
V
2
n
− DU
2
n
=4Q
n

for every n ∈ Z.
This may also be put in the form:
U
2
n+1
− PU
n+1
U
n
+ QU
2
n
= Q
n
.
(2.7) Conversion formulas:
DU
n
= V
n+1
− QV
n−1
,
V
n
= U
n+1
− QU
n−1
,

for every n ∈ Z.
2 Basic properties 9
(2.8) Addition of indices:
U
m+n
= U
m
V
n
− Q
n
U
m−n
,
V
m+n
= V
m
V
n
− Q
n
V
m−n
= DU
m
U
n
+ Q
n

V
m−n
,
for all m, n ∈ Z.
Other formulas of the same kind are:
2U
m+n
= U
m
V
n
+ U
n
V
m
,
2Q
n
U
m−n
= U
m
V
n
− U
n
V
m
,
for all m, n ∈ Z.

(2.9) Multiplication of indices:
U
2n
= U
n
V
n
,
V
2n
= V
2
n
− 2Q
n
,
U
3n
= U
n
(V
2
n
− Q
n
)=U
n
(DU
2
n

+3Q
n
),
V
3n
= V
n
(V
2
n
− 3Q
n
),
for every n ∈ Z.
More generally, if k ≥ 3 it is possible to find by induction on k
formulas for U
kn
and V
kn
, but I shall refrain from giving them
explicitly.
E. Divisibility properties
(2.10) Let U
m
= 1. Then, U
m
divides U
n
if and only if m | n.
Let V

m
= 1. Then, V
m
divides V
n
if and only if m | n and n/m is
odd.
For the next properties, it will be assumed that gcd(P, Q)=1.
(2.11) gcd(U
m
,U
n
)=U
d
,whered =gcd(m, n).
(2.12)
gcd(V
m
,V
n
)=



V
d
if
m
d
and

n
d
are odd,
1 or 2 otherwise,
where d = gcd(m, n).
10 1. The Fibonacci Numbers and the Arctic Ocean
(2.13)
gcd(U
m
,V
n
)=



V
d
if
m
d
is even,
n
d
is odd,
1 or 2 otherwise,
where d = gcd(m, n).
(2.14) If n ≥ 1, then gcd(U
n
,Q) = 1 and gcd(V
n

,Q)=1.
3 Prime divisors of Lucas sequences
The classical results about prime divisors of terms of Lucas se-
quences date back to Euler, (for numbers
a
n
−b
n
a−b
), to Lucas (for
Fibonacci and Lucas numbers), and to Carmichael (for other Lucas
sequences).
A. The sets P(U ), P(V ), and the rank of appearance.
Let P denote the set of all prime numbers. Given the Lucas sequences
U =(U
n
(P, Q))
n≥0
, V =(V
n
(P, Q))
n≥0
,let
P(U)={p ∈P|∃n ≥ 1 such that U
n
= 0 and p | U
n
},
P(V )={p ∈P|∃n ≥ 1 such that V
n

= 0 and p | V
n
}.
If U, V are degenerate, then P(U), P(V ) are easily determined sets.
Therefore, it will be assumed henceforth that U, V are non-
degenerate and thus, U
n
(P, Q) =0,V
n
(P, Q) = 0 for all n ≥
1.
Note that if p is a prime dividing both p, q,thenp | U
n
(P, Q),
p | V
n
(P, Q), for all n ≥ 2. So, for the considerations which will
follow, there is no harm in assuming that gcd(P, Q)=1.So,(P,Q)
belongs to the set
S = {(P, Q) | P ≥ 1, gcd(P, Q)=1,P
2
= Q, 2Q, 3Q, 4Q}.
For each prime p, define
ρ
U
(p)=

n if n is the smallest positive index where p | U
n
,

∞ if p  U
n
for every n>0,
ρ
V
(p)=

n if n is the smallest positive index where p | V
n
,
∞ if p  V
n
for every n>0.
3 Prime divisors of Lucas sequences 11
We call ρ
U
(n) (respectively ρ
V
(p))) is called the rank of appearance
of p in the Lucas sequence U (respectively V ).
First, I consider the determination of even numbers in the Lucas
sequences.
(3.1) Let n ≥ 0. Then:
U
n
even ⇐⇒








P even Q odd, n even,
or
P odd Q odd, 3 | n,
and
V
n
even ⇐⇒







P even Q odd, n ≥ 0,
or
P odd Q odd, 3 | n.
Special Cases. For the sequences of Fibonacci and Lucas numbers
(P =1,Q = −1), one has:
U
n
is even if and only if 3 | n,
V
n
is even if and only if 3 | n.
For the sequences of numbers U
n

=
a
n
−b
n
a−b
, V
n
= a
n
+ b
n
,witha>
b ≥ 1, gcd(a, b)= 1,p = a + b, q = ab, one has:
If a, b are odd, then U
n
is even if and only if n is even, while V
n
is
even for every n.
If a, b have different parity, then U
n
, V
n
are always odd (for n ≥ 1).
With the notations and terminology introduced above the result
(3.1) may be rephrased in the following way:
(3.2) 2 ∈P(U) if and only if Q is odd
ρ
U

(2) =







2ifP even, Q odd,
3ifP odd, Q odd,
∞ if P odd, Q even,
2 ∈P(V ) if and only if Q is odd
ρ
V
(2) =







1ifP even, Q odd,
3ifP odd, Q odd,
∞ if P odd, Q even.
12 1. The Fibonacci Numbers and the Arctic Ocean
Moreover, if Q is odd, then 2 | U
n
(respectively 2 | V
n

) if and only if
ρ
U
(2) | n (respectively ρ
V
(2) | n).
This last result extends to odd primes:
(3.3) Let p be an odd prime.
If p ∈P(U ), then p | U
n
if and only if ρ
U
(p) | n.
If p ∈P(V ), then p | V
n
if and only if ρ
V
(p) | n and
n
ρ
V
(p)
is odd.
Now I consider odd primes p and indicate when p ∈P(U).
(3.4) Let p be an odd prime.
If p  P and p | Q,thenp  U
n
for every n ≥ 1.
If p | P and p  Q,thenp | U
n

if and only if n is even.
If p  PQ and p | D,thenp | U
n
if and only if p | n.
If p  PQD,thenp divides U
ψ
D
(p)whereψ
D
(p)=p − (
D
p
)and(
D
p
)
denotes the Legendre symbol.
Thus,
P(U)={p ∈P|p  Q},
so P(U) is an infinite set.
The more interesting assertion concerns the case where p  PQD,
the other ones being very easy to establish.
The result may be expressed in terms of the rank of appearance:
(3.5) Let p be an odd prime.
If p  P, p | Q,thenρ
U
(p)=∞.
If p | P, p  Q,thenρ
U
(p)=2.

If p  PQ, p | D,thenρ
U
(p)=p.
If p  PQD,thenρ
U
(p) | Ψ
D
(p).
Special Cases. For the sequences of Fibonacci numbers (P =1,
Q = −1), D =5and5| U
n
if and only if 5 | n.
If p is an odd prime, p =5,thenp | U
p−(
5
p
)
,soρ
U
(p) | (p − (
5
p
)).
Because U
3
= 2, it follows that P(U )=P.
Let a>b≥ 1, gcd(a, b), P = a + b, Q = ab, U
n
=
a

n
−b
n
a−b
.
If p divides a or b but not both a, b,thenp  U
n
for all n ≥ 1.
If p  ab, p | a + b,thenp | U
n
if and only if n is even.
If p  ab(a + b) but p | a − b,thenp | U
n
if and only if p | n.
If p  ab(a + b)(a −b), then p | U
p−1
. (Note that D =(a − b)
2
).
Thus, P(U)={p : p  ab}.
3 Prime divisors of Lucas sequences 13
Taking b =1,ifp  a,thenp | U
p−1
, hence p | a
p−1
− 1 (this is
Fermat’s Little Theorem, which is therefore a special case of the last
assertion of (3.4)); it is trivial if p | (a +1)(a − 1).
The result (3.4) is completed with the so-called law of repetition,
first discovered by Lucas for the Fibonacci numbers:

(3.6) Let p
e
(with e ≥ 1) be the exact power of p dividing U
n
.Let
f ≥ 1, p  k. Then, p
e+f
divides U
nkp
f
.Moreover,ifp  Q, p
e
=2,
then p
e+f
is the exact power of p dividing U
nkp
e
.
It was seen above that Fermat’s Little Theorem is a special case of
the assertion that if p is a prime and p  PQD,thenp divides U
Ψ
D
(p)
.
I indicate now how to reinterpret Euler’s classical theorem.
If α, β are the roots of the characteristic polynomial X
2
−PX+Q,
define the symbol


α, β
2

=







1ifQ is even,
0ifQ is odd, P is even,
−1ifQ is odd, P is odd,
and for any odd prime p

α, β
p

=






D
p


if p  D,
0ifp | D.
Let Ψ
α,β
(p)=p − (
α,β
p
) for every prime p. Thus, using the previous
notation, Ψ
α,β
(p)=Ψ
D
(p)whenp is odd and p  D.
For n =

p
p
e
, define the generalized Euler function
Ψ
α,β
(n)=n

r
Ψ
α,β
(p)
p
,
so Ψ

α,β
(p
e
)=p
e−1
Ψ
α,β
(p) for each prime p and e ≥ 1. Define also
the Carmichael function λ
α,β
(n)=lcm{Ψ
α,β
(p
e
)}.Thus,λ
α,β
(n)
divides Ψ
α,β
(n).
In the special case where α = a, β =1,anda is an integer,
then Ψ
a,1
(p)=p − 1 for each prime p not dividing a. Hence, if
gcd(a, n) = 1, then Ψ
a,1
(n)=ϕ(n), where ϕ denotes the classical
Euler function.
The generalization of Euler’s theorem by Carmichael is the
following:

14 1. The Fibonacci Numbers and the Arctic Ocean
(3.7) n divides U
λ
α,β
(n)
hence, also, U
Ψ
α,β
(n)
.
It is an interesting question to evaluate the quotient
Ψ
D
(p)
ρ
U
(p)
.Itwas
shown by Jarden (1958) that for the sequence of Fibonacci numbers,
sup

p − (
5
p
)
ρ
U
(D)

= ∞

(as p tends to ∞). More generally, Kiss (1978) showed:
(3.8) (a) For each Lucas sequence U
n
(P, Q),
sup

Ψ
D
(p)
ρ
U
(p)

= ∞.
(b) There exists C>0 (depending on P , Q) such that
Ψ
D
(p)
ρ
U
(p)
<C
p
log p
.
Now I turn my attention to the companion Lucas sequence V =
(V
n
(P, Q))
n≥0

and I study the set of primes P(V ). It is not known
how to describe explicitly, by means of finitely many congruences,
the set P(V ). I shall indicate partial congruence conditions that are
complemented by density results.
Because U
2n
= U
n
V
n
, it then follows that P(V ) ⊆P(U). It was
already stated that 2 = P(V ) if and only if Q is odd.
(3.9) Let p be an odd prime.
If p  P, p | Q,thenp  V
n
for all n ≥ 1.
If p | P, p  Q,thenp | V
n
if and only if n is odd.
If p  PQ, p | D,thenp  V
n
for all n ≥ 1.
If p  PQD,thenp | V
1
2
Ψ
D
(p)
if and only if (
Q

P
)=−1.
If p  PQD and (
Q
p
) = 1, (
D
p
)=−(
−1
p
), then p  V
n
for all n ≥ 1.
The above result implies that P(V ) is an infinite set.
∗
One may fur-
ther refine the last two assertions; however, a complete determination
of P(V ) is not known.
In terms of the rank of appearance, (3.9) can be rephrased as
follows:
∗
This was extended by Ward (1954) for all binary linear recurrences
3 Prime divisors of Lucas sequences 15
(3.10) Let p be an odd prime.
If p | P, p  Q,thenρ
V
(p)=1.
If p  P, p | Q,thenρ
V

(p)=∞.
If p  PQ, p | D,thenρ
V
(p)=∞.
If p  PQD,(
Q
p
)=−1, then ρ
V
(p) divides
1
2
Ψ
D
(p).
If p  PQD,(
Q
p
) = 1, (
D
p
)=−(
−1
p
), then ρ
V
(p)=∞.
The following conjecture has not yet been established in general,
but has been verified in special cases, described below:
Conjecture. For each companion Lucas sequence V , the limit

δ(V ) = lim
π
V
(x)
π(x)
exists and is strictly greater than 0.
Here, π(x)=#{p ∈P|p ≤ x} and π
V
(x)=#{p ∈P(V ) | p ≤ x}.
The limit δ(V ) is the density of the set of prime divisors of V among
all primes.
Special Cases. Let (P, Q)=(1, −1), so V is the sequence of Lucas
numbers. Then the above results may be somewhat completed. Ex-
plicitly:
If p ≡ 3, 7, 11, 19 (mod 20), then p ∈P(V ).
If p ≡ 13, 17 (mod 20), then p/∈P(V ).
If p ≡ 1, 9 (mod 20) it may happen that p ∈P(V )orthatp/∈P(V ).
Jarden (1958) showed that there exist infinitely many primes p ≡
1 (mod 20) in P(V ) and also infinitely many primes p ≡ 1 (mod 20)
not in P(V ). Further results were obtained by Wa rd (1961) who
concluded that there is no finite set of congruences to decide if an
arbitrary prime p is in P(V ).
Inspired by a method of Hasse (1966), and the analysis of Ward
(1961), Lagarias (1985) showed that, for the sequence V of Lucas
numbers, the density is δ(V )=
2
3
.
Brauer (1960) and Hasse (1966) studied a problem of Sierpi
´

n-
ski, namely, determine the primes p such that 2 has an even order
modulo p, equivalently, determine the primes p dividing the numbers
2
n
+1 = V
n
(3, 2). He proved that δ(V (3, 2)) = 17/24. Lagarias
pointed out that Hasse’s proof shows also that if a ≥ 3issquare-
free, then δ(V (a +1,a)) = 2/3; see also a related paper of Hasse
(1965).