My Numbers, My Friends: Popular Lectures on Number Theory

Size: px
Start display at page:

Download "My Numbers, My Friends: Popular Lectures on Number Theory"

Transcription

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).

The square-free kernel of x 2n a 2n

The square-free kernel of x 2n a 2n ACTA ARITHMETICA 101.2 (2002) The square-free kernel of x 2n a 2n by Paulo Ribenboim (Kingston, Ont.) Dedicated to my long-time friend and collaborator Wayne McDaniel, at the occasion of his retirement

More information

Prime Numbers. Chapter Primes and Composites

Prime Numbers. Chapter Primes and Composites Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Die ganzen zahlen hat Gott gemacht

Die ganzen zahlen hat Gott gemacht Die ganzen zahlen hat Gott gemacht Polynomials with integer values B.Sury A quote attributed to the famous mathematician L.Kronecker is Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

More information

11 Ideals. 11.1 Revisiting Z

11 Ideals. 11.1 Revisiting Z 11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

More information

Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

More information

The cyclotomic polynomials

The cyclotomic polynomials The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =

More information

Continued Fractions. Darren C. Collins

Continued Fractions. Darren C. Collins Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

More information

k, then n = p2α 1 1 pα k

k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

More information

Primality - Factorization

Primality - Factorization Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

More information

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

More information

Prime and Composite Terms in Sloane s Sequence A056542

Prime and Composite Terms in Sloane s Sequence A056542 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.3 Prime and Composite Terms in Sloane s Sequence A056542 Tom Müller Institute for Cusanus-Research University and Theological

More information

ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

More information

On Generalized Fermat Numbers 3 2n +1

On Generalized Fermat Numbers 3 2n +1 Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic

More information

Primitive Prime Divisors of First-Order Polynomial Recurrence Sequences

Primitive Prime Divisors of First-Order Polynomial Recurrence Sequences Primitive Prime Divisors of First-Order Polynomial Recurrence Sequences Brian Rice brice@hmc.edu July 19, 2006 Abstract The question of which terms of a recurrence sequence fail to have primitive prime

More information

Homework until Test #2

Homework until Test #2 MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

More information

FACTORING SPARSE POLYNOMIALS

FACTORING SPARSE POLYNOMIALS FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

More information

On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

More information

PRIME FACTORS OF CONSECUTIVE INTEGERS

PRIME FACTORS OF CONSECUTIVE INTEGERS PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

More information

MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

More information

Factoring & Primality

Factoring & Primality Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount

More information

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear: MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

More information

PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

More information

Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

More information

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

More information

Row Ideals and Fibers of Morphisms

Row Ideals and Fibers of Morphisms Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

More information

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

The van Hoeij Algorithm for Factoring Polynomials

The van Hoeij Algorithm for Factoring Polynomials The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

More information

CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

More information

Chapter Two. Number Theory

Chapter Two. Number Theory Chapter Two Number Theory 2.1 INTRODUCTION Number theory is that area of mathematics dealing with the properties of the integers under the ordinary operations of addition, subtraction, multiplication and

More information

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12 CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

More information

Handout NUMBER THEORY

Handout NUMBER THEORY Handout of NUMBER THEORY by Kus Prihantoso Krisnawan MATHEMATICS DEPARTMENT FACULTY OF MATHEMATICS AND NATURAL SCIENCES YOGYAKARTA STATE UNIVERSITY 2012 Contents Contents i 1 Some Preliminary Considerations

More information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.

EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p. EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If

More information

ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS

ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number

More information

Lecture 3. Mathematical Induction

Lecture 3. Mathematical Induction Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

More information

Characterizing the Sum of Two Cubes

Characterizing the Sum of Two Cubes 1 3 47 6 3 11 Journal of Integer Sequences, Vol. 6 (003), Article 03.4.6 Characterizing the Sum of Two Cubes Kevin A. Broughan University of Waikato Hamilton 001 New Zealand kab@waikato.ac.nz Abstract

More information

A Course on Number Theory. Peter J. Cameron

A Course on Number Theory. Peter J. Cameron A Course on Number Theory Peter J. Cameron ii Preface These are the notes of the course MTH6128, Number Theory, which I taught at Queen Mary, University of London, in the spring semester of 2009. There

More information

NUMBER THEORY AMIN WITNO

NUMBER THEORY AMIN WITNO NUMBER THEORY AMIN WITNO ii Number Theory Amin Witno Department of Basic Sciences Philadelphia University JORDAN 19392 Originally written for Math 313 students at Philadelphia University in Jordan, this

More information

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

On the largest prime factor of the Mersenne numbers

On the largest prime factor of the Mersenne numbers On the largest prime factor of the Mersenne numbers Kevin Ford Department of Mathematics The University of Illinois at Urbana-Champaign Urbana Champaign, IL 61801, USA ford@math.uiuc.edu Florian Luca Instituto

More information

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM Acta Math. Univ. Comenianae Vol. LXXXI, (01), pp. 03 09 03 ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

THE CONGRUENT NUMBER PROBLEM

THE CONGRUENT NUMBER PROBLEM THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean

More information

Solutions to Practice Problems

Solutions to Practice Problems Solutions to Practice Problems March 205. Given n = pq and φ(n = (p (q, we find p and q as the roots of the quadratic equation (x p(x q = x 2 (n φ(n + x + n = 0. The roots are p, q = 2[ n φ(n+ ± (n φ(n+2

More information

Is n a prime number? Nitin Saxena. Turku, May Centrum voor Wiskunde en Informatica Amsterdam

Is n a prime number? Nitin Saxena. Turku, May Centrum voor Wiskunde en Informatica Amsterdam Is n a prime number? Nitin Saxena Centrum voor Wiskunde en Informatica Amsterdam Turku, May 2007 Nitin Saxena (CWI, Amsterdam) Is n a prime number? Turku, May 2007 1 / 36 Outline 1 The Problem 2 The High

More information

SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

More information

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

More information

2 The Euclidean algorithm

2 The Euclidean algorithm 2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

More information

OSTROWSKI FOR NUMBER FIELDS

OSTROWSKI FOR NUMBER FIELDS OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for

More information

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

More information

The Mathematics of Origami

The Mathematics of Origami The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

More information

RESULTANT AND DISCRIMINANT OF POLYNOMIALS

RESULTANT AND DISCRIMINANT OF POLYNOMIALS RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results

More information

Stanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D.

Stanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D. Stanford University Educational Program for Gifted Youth (EPGY) Dana Paquin, Ph.D. paquin@math.stanford.edu Summer 2010 Note: These lecture notes are adapted from the following sources: 1. Ivan Niven,

More information

On the number-theoretic functions ν(n) and Ω(n)

On the number-theoretic functions ν(n) and Ω(n) ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,

More information

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

More information

Practice Problems for First Test

Practice Problems for First Test Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

More information

Two-generator numerical semigroups and Fermat and Mersenne numbers

Two-generator numerical semigroups and Fermat and Mersenne numbers Two-generator numerical semigroups and Fermat and Mersenne numbers Shalom Eliahou and Jorge Ramírez Alfonsín Abstract Given g N, what is the number of numerical semigroups S = a,b in N of genus N\S = g?

More information

Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

More information

Series. Chapter Convergence of series

Series. Chapter Convergence of series Chapter 4 Series Divergent series are the devil, and it is a shame to base on them any demonstration whatsoever. Niels Henrik Abel, 826 This series is divergent, therefore we may be able to do something

More information

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

More information

Solving a family of quartic Thue inequalities using continued fractions

Solving a family of quartic Thue inequalities using continued fractions Solving a family of quartic Thue inequalities using continued fractions Andrej Dujella, Bernadin Ibrahimpašić and Borka Jadrijević Abstract In this paper we find all primitive solutions of the Thue inequality

More information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

More information

Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

More information

8 Divisibility and prime numbers

8 Divisibility and prime numbers 8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

More information

1. R In this and the next section we are going to study the properties of sequences of real numbers.

1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

More information

Settling a Question about Pythagorean Triples

Settling a Question about Pythagorean Triples Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:

More information

The last three chapters introduced three major proof techniques: direct,

The last three chapters introduced three major proof techniques: direct, CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

More information

QUADRATIC RECIPROCITY IN CHARACTERISTIC 2

QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 KEITH CONRAD 1. Introduction Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] (see [4, Section 3.2.2] or [5]) lets

More information

Algebraic and Transcendental Numbers

Algebraic and Transcendental Numbers Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

MATH 361: NUMBER THEORY FIRST LECTURE

MATH 361: NUMBER THEORY FIRST LECTURE MATH 361: NUMBER THEORY FIRST LECTURE 1. Introduction As a provisional definition, view number theory as the study of the properties of the positive integers, Z + = {1, 2, 3, }. Of particular interest,

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

More information

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

3. QUADRATIC CONGRUENCES

3. QUADRATIC CONGRUENCES 3. QUADRATIC CONGRUENCES 3.1. Quadratics Over a Finite Field We re all familiar with the quadratic equation in the context of real or complex numbers. The formula for the solutions to ax + bx + c = 0 (where

More information

Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com

Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

Mathematical Induction

Mathematical Induction Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

More information

SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

More information

7. Some irreducible polynomials

7. Some irreducible polynomials 7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

More information