My Numbers, My Friends: Popular Lectures on Number Theory

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1 My Numbers, My Friends: Popular Lectures on Number Theory Paulo Ribenboim Springer

2 My Numbers, My Friends

3

4 Paulo Ribenboim My Numbers, My Friends Popular Lectures on Number Theory

5 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 (sc. : alk. paper) 1. Number Theory. I. Title QA241.R dc 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 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 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 Springer-Verlag New York Berlin Heidelberg SPIN

6 Contents Preface xi 1 The Fibonacci Numbers and the Arctic Ocean 1 1 Basicdefinitions A. Lucassequences B. SpecialLucassequences C. Generalizations Basicproperties A. Binet sformulas B. DegenerateLucassequences C. Growth and numerical calculations D. Algebraicrelations E. Divisibility properties Prime divisors of Lucas sequences A. The sets P(U), P(V ), and the rank of appearance B. Primitive factors of Lucas sequences PrimesinLucassequences Powers and powerful numbers in Lucas sequences. 28 A. Generaltheoremsforpowers B. Explicit determination in special sequences. 30

7 vi Contents C. Uniform explicit determination of multiples, squares, and square-classes for certain families of Lucas sequences D. Powerful numbers in Lucas sequences 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 Ringsofintegers Discriminant Decompositionofprimes A. Propertiesofthenorm Units Theclassnumber A. Calculationoftheclassnumber B. Determination of all quadratic fields with classnumber Themaintheorem Gauss and the Class Number Problem Introduction HighlightsofGauss life Briefhistoricalbackground Binaryquadraticforms The fundamental problems Equivalenceofforms Conditional solution of the fundamental problems Proper equivalence classes of definite forms A. Anothernumericalexample Proper equivalence classes of indefinite forms A. Anothernumericalexample Theautomorphofaprimitiveform Composition of proper equivalence classes of primitiveforms

8 Contents vii 12 Thetheoryofgenera The group of proper equivalence classes of primitiveforms Calculationsandconjectures The aftermath of Gauss (or the math after Gauss) Formsversusidealsinquadraticfields Dirichlet sclassnumberformula Solution of the class number problem for definite forms The class number problem for indefinite forms Morequestionsandconjectures Many topics have not been discussed Consecutive Powers Introduction History Specialcases Divisibility properties Estimates A. The equation a U b V = B. The equation X m Y n = C. The equation X U Y V = Final comments and applications A. Determination of the residue of q p (a) B. Identities and congruences for the Fermat quotient Powerless Facing Powers Powerfulnumbers A. Distributionofpowerfulnumbers B. Additiveproblems C. Differenceproblems Powers A. Pythagorean triples and Fermat s problem. 235 B. VariantsofFermat sproblem C. TheconjectureofEuler D. The equation AX l + BY m = CZ n

9 viii Contents E. Powersasvaluesofpolynomials Exponentialcongruences A. TheWieferichcongruence B. Primitivefactors Dreammathematics A. The statements B. Statements C. Binomials and Wieferich congruences D. Erdös conjecture and Wieferich congruence. 257 E. Thedreaminthedream What Kind of Number Is 2 2? Introduction Kindsofnumbers Hownumbersaregiven Briefhistoricalsurvey Continuedfractions A. Generalities B. Periodiccontinuedfractions C. Simple continued fractions of π and e Approximationbyrationalnumbers A. Theorderofapproximation B. TheMarkoffnumbers C. Measuresofirrationality D. Order of approximation of irrational algebraicnumbers Irrationalityofspecialnumbers Transcendental numbers A. Liouville numbers B. Approximation by rational numbers: sharpertheorems C. Hermite, Lindemann, and Weierstrass D. AresultofSiegelonexponentials E. Hilbert s7thproblem F. TheworkofBaker G. TheconjectureofSchanuel H. Transcendence measure and the classificationofmahler Final comments

10 Contents ix 11 Galimatias Arithmeticae 344 Index of Names 361 Index of Subjects 369

11 Preface Dear Friends of Numbers: This little book is for you. It should offer an exquisite intellectual 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 being 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.

12 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 rabbits. The first significant work on the subject is by Lucas, with his seminal paper of 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 sequences 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.

13 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 2 PX 1 PX + QX 2 = n=0 U n X n V n X n. and 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

14 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 consecutive 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 important 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 = an 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,

15 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 extended 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 Let W 0 = PT 0 +2T 1 and W 1 =2QT 0 + PT 1. 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, normalized, 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 α n β n α 2 β 2 if n is odd, 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 thatd = 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

16 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 = αn 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) V n ( P, Q) =( 1) n V n (P, Q). and 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.

17 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 ( ) P Q M =. 1 0 Then for n 1, ( Un ) ) U n 1 = M n 1 ( 1 0

18 2 Basic properties 7 and ( ) ( Vn = M n 1 2 V n 1 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 2e 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 k e 2 e, where k i = 0 or 1, then M k = M k 0 (M 2 ) k 1... (M 2e ) ke. 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 αn < 1 D 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 αn 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 = , (the golden number), β = (1 5)/2 = 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 endeavor 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.

19 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, ( ) ( ) n 2 n 3 U n = P n 1 P n 3 Q + P n 5 Q ( ) n 1 k +( 1) k P n 1 2k Q k + + (last summand) k 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: for every n Z. V 2 n DU 2 n =4Q n This may also be put in the form: (2.7) Conversion formulas: for every n Z. U 2 n+1 PU n+1 U n + QU 2 n = Q n. DU n = V n+1 QV n 1, V n = U n+1 QU n 1,

20 2 Basic properties 9 (2.8) Addition of indices: for all m, n Z. 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, Other formulas of the same kind are: for all m, n Z. (2.9) Multiplication of indices: for every n Z. 2U m+n = U m V n + U n V m, 2Q n U m n = U m V n U n V m, 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 ), 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) V d if m gcd(v m,v n )= d and n d 1 or 2 otherwise, are odd, where d = gcd(m, n).

21 10 1. The Fibonacci Numbers and the Arctic Ocean (2.13) V d if m gcd(u m,v n )= d is even, n d 1 or 2 otherwise, is odd, 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 sequences date back to Euler, (for numbers an 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 andp U n }, P(V )={p P n 1 such that V n = 0 andp 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 nondegenerate 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 { n if n is the smallest positive index where p Un, ρ U (p) = if p U n for every n>0, { n if n is the smallest positive index where p Vn, ρ V (p) = if p V n for every n>0.

22 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: P even Q odd, n even, U n even or P odd Q odd, 3 n, and P even Q odd, n 0, V n even 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 = an 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 2 if P even, Q odd, ρ U (2) = 3 if P odd, Q odd, if P odd, Q even, 2 P(V ) if and only if Q is odd 1 if P even, Q odd, ρ V (2) = 3 if P odd, Q odd, if P odd, Q even.

23 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) Now I consider odd primes p and indicate when p P(U). is odd. (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 ψd (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 ),soρ U(p) (p ( 5 p 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 = an 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}.

24 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 ΨD (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 1 if Q is even, = 0 if Q is odd, P is even, 2 1 if Q is odd, P is odd, and for any odd prime p ( ) ( ) α, β D if p D, = p p 0 if p 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 pe, 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, and a 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:

25 14 1. The Fibonacci Numbers and the Arctic Ocean (3.7) n divides U λα,β (n) hence, also, U Ψα,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, { p ( 5 p sup ) } = ρ U (D) (as p tends to ). More generally, Kiss (1978) showed: (3.8) (a) For each Lucas sequence U n (P, Q), { } ΨD (p) sup =. 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 1 p )= ( p ), then p V n for all n 1. The above result implies that P(V ) is an infinite set. One may further 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

26 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. Explicitly: 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 Ward (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 Sierpiński, 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 3issquarefree, then δ(v (a +1,a)) = 2/3; see also a related paper of Hasse (1965).

27 16 1. The Fibonacci Numbers and the Arctic Ocean Laxton (1969) considered, for each a 2, the set W(a) ofall binary linear recurrences W with W 0, W 1 satisfying W 1 =W 0, W 1 = aw 0,andW n =(a +1)W n 1 aw n 2,forn 2. This set includes the Lucas sequences U(a +1,a), V (a +1,a). For each prime p, let e p (a) = { 0 if p a, order of a mod b if p a. Laxton gave a heuristic argument to the effect that if the limit, as x tends to, of 1 e p (a) π(x) p 1 p x exists, then it is the expected (or average value), for any W W(a), of the density of primes in P(W) (that is, the set of primes dividing some W n ). Stephens (1976) used a method of Hooley (1967) who had proved, under the assumption of a generalized Riemann s hypothesis, Artin s conjecture that 2 is a primitive root modulo p for infinitely many primes p. Leta 2, a not a proper power. Assume the generalized Riemann hypothesis for the Dedekind ζ function of all fields Q(a 1/n,ζ k ), where ζ k is a primitive kth root of 1. Then, for every x 2, ( ) e p (a) p 1 = c(a) x x log log x log x + O (log x) 2 ; p x by the Prime Number Theorem, the limit considered above exists and is equal to c(a). Stephens evaluated c(a). Let C = ( 1 p ) p p 3, 1 let a = a 1 (a 2 ) 2 where a 1 is square-free, let r be the number of distinct prime factors of a 1, and let f be defined as 2 5 f = 1 64 if a 1 1(mod4), if a 1 2(mod4), 1 20 if a 1 3(mod4).

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