Thomas koshy elementary number theory with applications academic press

Elementary Number Theory withApplications Second Edition... Elementary Number Theory withApplications Second Edition Thomas Koshy Academic Press is an imprint of Elsevier... Elementary n

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List of Symbols

( a1, a2, , a n) the greatest common factor of a1, a2, , and a n (162)

[a1, a2, , a n] the least common multiple of a1, a2, , and a n (187)

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Elementary Number Theory with

Applications Second Edition

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Elementary Number Theory with

Applications Second Edition

Thomas Koshy

Academic Press is an imprint of Elsevier

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30 Corporate Drive, Suite 400, Burlington, MA 01803, USA

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Library of Congress Cataloging-in-Publication Data

Koshy, Thomas.

Elementary number theory with applications / Thomas Koshy – 2nd ed.

p cm.

Includes bibliographical references and index.

ISBN 978-0-12-372487-8 (alk paper)

1 Number theory I Title.

QA241.K67 2007

512.7–dc22

2007010165

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

ISBN: 978-0-12-372487-8

For information on all Academic Press publications

visit our Web site at www.books.elsevier.com

Printed in the United States of America

07 08 09 10 9 8 7 6 5 4 3 2 1

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M K Tharian; and to the memory of Professor Edwin Weiss, Professor Donald W Blackett, and Vice Chancellor A V Varughese

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Preface xiii

A Word to the Student xxi

1 Fundamentals 1

1.1 Fundamental Properties 3

1.2 The Summation and Product Notations 9

1.3 Mathematical Induction 15

1.4 Recursion 26

1.5 The Binomial Theorem 32

1.6 Polygonal Numbers 39

1.7 Pyramidal Numbers 49

1.8 Catalan Numbers 52

Chapter Summary 57

Review Exercises 60

Supplementary Exercises 62

Computer Exercises 65

Enrichment Readings 66

2 Divisibility 69

2.1 The Division Algorithm 69

2.2 Base-b Representations (optional) 80

2.3 Operations in Nondecimal Bases (optional) 89

2.4 Number Patterns 98

2.5 Prime and Composite Numbers 103

2.6 Fibonacci and Lucas Numbers 128

2.7 Fermat Numbers 139

Chapter Summary 143

Review Exercises 146

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3 Greatest Common Divisors 155

3.1 Greatest Common Divisor 155

3.2 The Euclidean Algorithm 166

3.3 The Fundamental Theorem of Arithmetic 173

3.4 Least Common Multiple 184

3.5 Linear Diophantine Equations 188

Chapter Summary 205

4 Congruences 211

4.1 Congruences 211

4.2 Linear Congruences 230

4.3 The Pollard Rho Factoring Method 238

Chapter Summary 240

5 Congruence Applications 247

5.1 Divisibility Tests 247

5.2 Modular Designs 253

5.3 Check Digits 259

5.4 The p-Queens Puzzle (optional) 273

5.5 Round-Robin Tournaments (optional) 277

5.6 The Perpetual Calendar (optional) 282

Chapter Summary 288

6 Systems of Linear Congruences 295

6.1 The Chinese Remainder Theorem 295

6.2 General Linear Systems (optional) 303

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6.3 2×2Linear Systems (optional) 307

Chapter Summary 313

7 Three Classical Milestones 321

7.1 Wilson’s Theorem 321

7.2 Fermat’s Little Theorem 326

7.3 Pseudoprimes (optional) 337

7.4 Euler’s Theorem 341

Chapter Summary 348

8 Multiplicative Functions 355

8.1 Euler’s Phi Function Revisited 355

8.2 The Tau and Sigma Functions 365

8.3 Perfect Numbers 373

8.4 Mersenne Primes 381

8.5 The Möbius Function (optional) 398

Chapter Summary 406

9 Cryptology 413

9.1 Affine Ciphers 416

9.2 Hill Ciphers 425

9.3 Exponentiation Ciphers 430

9.4 The RSA Cryptosystem 434

9.5 Knapsack Ciphers 443

Chapter Summary 448

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10 Primitive Roots and Indices 455

10.1 The Order of a Positive Integer 455

10.2 Primality Tests 464

10.3 Primitive Roots for Primes 467

10.4 Composites with Primitive Roots (optional) 474

10.5 The Algebra of Indices 482

Chapter Summary 489

11 Quadratic Congruences 495

11.1 Quadratic Residues 495

11.2 The Legendre Symbol 501

11.3 Quadratic Reciprocity 515

11.4 The Jacobi Symbol 527

11.5 Quadratic Congruences with Composite Moduli (optional) 535

Chapter Summary 543

12 Continued Fractions 551

12.1 Finite Continued Fractions 552

12.2 Infinite Continued Fractions 565

Chapter Summary 575

13 Miscellaneous Nonlinear Diophantine Equations 579

13.1 Pythagorean Triangles 579

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13.2 Fermat’s Last Theorem 590

13.3 Sums of Squares 602

13.4 Pell’s Equation 613

Chapter Summary 621

A Appendix 631

A.1 Proof Methods 631

A.2 Web Sites 638

T Tables 641

T.1 Factor Table 642

T.2 Values of Some Arithmetic Functions 649

T.3 Least Primitive Roots r Modulo Primes p 652

T.4 Indices 653

R References 657

S Solutions to Odd-Numbered Exercises 665

Chapter 1 Fundamentals 665

Chapter 2 Divisibility 677

Chapter 3 Greatest Common Divisors 689

Chapter 4 Congruences 696

Chapter 5 Congruence Applications 702

Chapter 6 Systems of Linear Congruences 707

Chapter 7 Three Classical Milestones 711

Chapter 8 Multiplicative Functions 718

Chapter 9 Cryptology 728

Chapter 10 Primitive Roots and Indices 731

Chapter 11 Quadratic Congruences 737

Chapter 12 Continued Fractions 746

Chapter 13 Miscellaneous Nonlinear Diophantine Equations 748

Credits 757

Index 761

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Man has the faculty of becoming completely absorbed in one subject,

no matter how trivial and no subject is so trivial that it will not assume infinite proportions if one’s entire attention is devoted to it.

— TOLSTOY , War and Peace

or over two thousand years, number theory has fascinated and inspired both

Famateurs and mathematicians alike A sound and fundamental body of

knowl-edge, it has been developed by the untiring pursuits of mathematicians allover the world Today, number theorists continue to develop some of the most so-phisticated mathematical tools ever devised and advance the frontiers of knowl-edge

Many number theorists, including the eminent nineteenth-century English ber theorist Godfrey H Hardy, once believed that number theory, although beautiful,had no practical relevance However, the advent of modern technology has brought

num-a new dimension to the power of number theory: constnum-ant prnum-acticnum-al use Once sidered the purest of pure mathematics, it is used increasingly in the rapid develop-ment of technology in a number of areas, such as art, coding theory, cryptology, andcomputer science The various fascinating applications have confirmed that humaningenuity and creativity are boundless, although many years of hard work may beneeded to produce more meaningful and delightful applications

con-The Pursuit of a Dream

This book is the fruit of years of dreams and the author’s fascination for the subject,its beauty, elegance, and historical development; the opportunities it provides forboth experimentation and exploration; and, of course, its marvelous applications.This new edition, building on the strengths of its predecessor, incorporates anumber of constructive suggestions made by students, reviewers, and well-wishers It

is logically conceived, self-contained, well-organized, nonintimidating, and writtenwith students and amateurs in mind In clear, readable language, this book offers anoverview of the historical development of the field, including major figures, as well

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as step-by-step development of the basic concepts and properties, leading to the moreadvanced exercises and discoveries.

Audience and Prerequisites

The book is designed for an undergraduate course in number theory for studentsmajoring in mathematics and/or computer science at the sophomore/junior level andfor students minoring in mathematics No formal prerequisites are required to studythe material or to enjoy its beauty except a strong background in college algebra.The main prerequisite is mathematical maturity: lots of patience, logical thinking,and the ability for symbolic manipulation This book should enable students andnumber theory enthusiasts to enjoy the material with great ease

Coverage

The text includes a detailed discussion of the traditional topics in an ate number theory course, emphasizing problem-solving techniques, applications,pattern recognition, conjecturing, recursion, proof techniques, and numeric compu-tations It also covers figurate numbers and their geometric representations, Catalannumbers, Fibonacci and Lucas numbers, Fermat numbers, an up-to-date discussion

undergradu-of the various classes undergradu-of prime numbers, and factoring techniques Starred ()

op-tional sections and opop-tional puzzles can be omitted without losing continuity of velopment

de-Included in this edition are new sections on Catalan numbers and the Pollard

rho factoring method, a subsection on the Pollard p− 1 factoring method, and ashort chapter on continued fractions The section on linear diophantine equationsnow appears in Chapter 3 to provide full prominence to congruences

A number of well-known conjectures have been added to challenge the more bitious students Identified by the conjecture symbol ? in the margin, they shouldprovide wonderful opportunities for group discussion, experimentation, and explo-ration

am-Examples and Exercises

Each section contains a wealth of carefully prepared and well-graded examples andexercises to enhance student skills Examples are developed in detail for easy un-derstanding Many exercise sets contain thought-provoking true/false problems, nu-meric problems to develop computational skills, and proofs to master facts and thevarious proof techniques Extensive chapter-end review exercise sets provide com-prehensive reviews, while chapter-end supplementary exercises provide challengingopportunities for the curious-minded to pursue

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Starred () exercises are, in general, difficult, and doubly starred () ones are

more difficult Both can be omitted without losing overall understanding of the cepts under discussion Exercises identified with a c in the margin require a knowl-edge of elementary calculus; they can be omitted by students with no calculus back-ground

con-Historical Comments and Biographies

Historical information, including biographical sketches of about 50 mathematicians,

is woven throughout the text to enhance a historical perspective on the ment of number theory This historical dimension provides a meaningful context forprospective and in-service high school and middle school teachers in mathematics

develop-An index of the biographies, keyed to pages in the text, can be found inside the backcover

Applications

This book has several unique features They include the numerous relevant andthought-provoking applications spread throughout, establishing a strong and mean-ingful bridge with geometry and computer science These applications increase stu-dent interest and understanding and generate student interaction In addition, thebook shows how modular systems can be used to create beautiful designs, link-ing number theory with both geometry and art The book also deals with barcodes,zip codes, International Serial Book Numbers, European Article Numbers, vehicleidentification numbers, and German bank notes, emphasizing the closeness of num-ber theory to our everyday life Furthermore, it features Friday-the-thirteenth, the

p-queens puzzle, round-robin tournaments, a perpetual calendar, the Pollard rho toring method, and the Pollard p− 1 factoring method

fac-Flexibility

The order and selection of topics offer maximum flexibility for instructors to selectchapters and sections that are appropriate for student needs and course lengths Forexample, Chapter 1 can be omitted or assigned as optional reading, as can the op-tional sections 6.2, 6.3, 7.3, 8.5, 10.4, and 11.5, without jeopardizing the core ofdevelopment Sections 2.2, 2.3, and 5.4–5.6 also can be omitted if necessary

Foundations

All proof methods are explained and illustrated in detail in the Appendix They vide a strong foundation in problem-solving techniques, algorithmic approach, andproof techniques

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Most concepts, definitions, and theorems are illustrated through thoughtfully selectedexamples Most of the theorems are proven, with the exception of some simple onesleft as routine exercises The proofs shed additional light on the understanding of thetopic and enable students to develop their problem-solving skills The various prooftechniques are illustrated throughout the text

Proofs Without Words

Several geometric proofs of formulas are presented without explanation This uniquefeature should generate class discussion and provide opportunities for further explo-ration

Pattern Recognition

An important problem-solving technique used by mathematicians is pattern tion Throughout the book, there are ample opportunities for experimentation and ex-ploration: collecting data, arranging them systematically, recognizing patterns, mak-ing conjectures, and then establishing or disproving these conjectures

recogni-Recursion

By drawing on well-selected examples, the text explains in detail this powerful egy, which is used heavily in both mathematics and computer science Many exam-ples are provided to ensure that students are comfortable with this powerful problem-solving technique

strat-Numeric Puzzles

Several fascinating, optional number-theoretic puzzles are presented for discussionand digression It would be a good exercise to justify each These puzzles are usefulfor prospective and in-service high school and middle school teachers in mathemat-ics

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Computer Assignments

Relevant and thought-provoking computer assignments are provided at the end ofeach chapter They provide hands-on experience with concepts and enhance the op-portunity for computational exploration and experimentation A computer algebra

system, such as Maple or Mathematica, or a language of your choice can be used.

Chapter Summary

At the end of each chapter, you will find a summary that is keyed to pages in the text.This provides a quick review and easy reference Summaries contain the variousdefinitions, symbols, and properties

The solutions to all odd-numbered exercises are given at the end of the text

Solutions Manual for Students

The Student’s Solutions Manual contains detailed solutions to all even-numbered

ex-ercises It also contains valuable tips for studying mathematics, as well as for ing and taking examinations

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prepar-Instructor’s Manual

The Instructor’s Manual contains detailed solutions to all even-numbered exercises,

sample tests for each chapter, and the keys for each test It also contains two samplefinal examinations and their keys

Highlights of this Edition

They include:

• Catalan numbers (Sections 1.8, 2.5, and 8.4)

• Linear diophantine equations with Fibonacci coefficients (Section 3.5)

• Pollard rho factoring method (Section 4.3)

• Vehicle identification numbers (Section 5.3)

• German bank notes (Section 5.3)

• Factors of 2n+ 1 (Section 7.2)

• Pollard p − 1 factoring method (Section 7.2)

• Pascal’s binary triangle and even perfect numbers (Section 8.4)

• Continued fractions (Chapter 12)

To begin with, I am indebted to the following reviewers for their boundless thusiasm and constructive suggestions:

en-Steven M Bairos Data Translation, Inc

Peter Brooksbank Bucknell UniversityRoger Cooke University of VermontJoyce Cutler Framingham State CollegeDaniel Drucker Wayne State UniversityMaureen Femick Minnesota State University at MankatoBurton Fein Oregon State University

Justin Wyss-Gallifent University of MarylandNapolean Gauthier The Royal Military College of CanadaRichard H Hudson University of South Carolina

Robert Jajcay Indiana State UniversityRoger W Leezer California State University at Sacramento

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I E Leonard University of Alberta

Seung H Son University of Colorado at Colorado Springs

Fernando Rodriguez Villegas University of Texas at Austin

Raymond E Whitney Lock Haven University

Thanks also to Roger Cooke of the University of Vermont, Daniel Drucker of WayneState University, Maureen Fenrick of Minnesota State University at Mankato, andKevin Jackson-Mead for combing through the entire manuscript for accuracy; toDaniel Drucker of Wayne State University and Dan Reich of Temple University forclass-testing the material; to the students Prasanth Kalakota of Indiana State Uni-versity and Elvis Gonzalez of Temple University for their comments; to Thomas E.Moore of Bridgewater State College and Don Redmond of Southern Illinois Uni-versity for preparing the solutions to all odd-numbered exercises; to Ward Heilman

of Bridgewater State College and Roger Leezer of California State University atSacramento for preparing the solutions to all even-numbered exercises; to MargariteRoumas for her superb editorial assistance; and to Madelyn Good and Ellen Keane atthe Framingham State College Library, who tracked down a number of articles andbooks My sincere appreciation also goes to Senior Editors Barbara Holland, whoinitiated the original project, Pamela Chester, and Thomas Singer; Production EditorChristie Jozwiak, Project Manager Jamey Stegmaier, Copyeditor Rachel Henriquez,and Editorial Assistant Karen Frost at Harcourt/Academic Press for their coopera-tion, promptness, support, encouragement, and confidence in the project

Finally, I must confess that any errors that may yet remain are my own sibility However, I would appreciate hearing about any inadvertent errors, alternatesolutions, or, better yet, exercises you have enjoyed

respon-Thomas Koshytkoshy@frc.mass.edu

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A Word to the Student

Mathematics is music for the mind; music is mathematics for the soul.

— ANONYMOUS

The Language of Mathematics

To learn a language, you have to know its alphabet, grammar, and syntax, and youhave to develop a decent vocabulary Likewise, mathematics is a language with itsown symbols, rules, terms, definitions, and theorems To be successful in mathe-matics, you must know them and be able to apply them; you must develop a work-ing vocabulary, use it as often as you can, and speak and write in the language ofmath

This book was written with you in mind, to create an introduction to numbertheory that is easy to understand Each chapter is divided into short sections of ap-proximately the same length

Problem-Solving Techniques

Throughout, the book emphasizes problem-solving techniques such as doing periments, collecting data, organizing them in an orderly fashion, recognizing pat-terns, and making conjectures It also emphasizes recursion, an extremely powerfulproblem-solving strategy used heavily in both mathematics and computer science.Although you may need some practice to get used to recursion, once you know how

ex-to approach problems recursively, you will appreciate its power and beauty So donot be turned off, even if you have to struggle a bit with it initially

The book stresses proof techniques as well Theorems are the bones of ematics So, for your convenience, the various proof methods are explained and il-lustrated in the Appendix It is strongly recommended that you master them; do theworked-out examples, and then do the exercises Keep reviewing the techniques asoften as needed

math-Many of the exercises use the theorems and the techniques employed in theirproofs Try to develop your own proofs This will test your logical thinking and

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analytical skills In order to fully enjoy this beautiful and elegant subject, you mustfeel at home with the various proof methods.

Getting Involved

Basketball players such as Michael Jordan and Larry Bird did not become stars by reading about basketball or watching others play Besides knowing the rulesand the objects needed to play, they needed countless hours of practice, hard work,and determination to achieve their goal Likewise, you cannot learn mathematics bysimply watching your professor do it in class or by reading about it; you have to

super-do it yourself every day, just as skill is acquired in a sport You can learn matics in small, progressive steps only, building on skills you already have devel-oped

mathe-Suggestions for Learning

Here are a few suggestions you should find useful in your pursuit:

• Read a few sections before each class You might not fully understand the terial, but you will certainly follow it far better when your professor discusses

ma-it in class Besides, you will be able to ask more questions in class and answermore questions

• Always go to class well prepared Be prepared to answer and ask questions

• Whenever you study from the book, make sure you have a pencil and enoughscrap paper next to you for writing the definitions, theorems, and proofs andfor doing the exercises

• Study the material taught in class on the same day Do not just read it as if youwere reading a novel or a newspaper Write down the definitions, theorems,and properties in your own words without looking in your notes or the book.Rewrite the examples, proofs, and exercises done in class, all in your ownwords If you cannot do them on your own, study them again and try again;continue until you succeed

• Always study the relevant section in the text and do the examples there, then

do the exercises at the end of the section Since the exercises are graded inorder of difficulty, do them in order Do not skip steps or write over previoussteps; this way you will be able to progress logically, locate your errors, andcorrect your mistakes If you cannot solve a problem because it involves anew term, formula, or some property, then re-study the relevant portion ofthe section and try again Do not assume that you will be able to do everyproblem the first time you try it Remember, practice is the best shortcut tosuccess

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Solutions Manual

The Student’s Solutions Manual contains additional tips for studying mathematics,

preparing for an examination in mathematics, and taking an examination in matics It also contains detailed solutions to all even-numbered exercises

mathe-A Final Word

Mathematics, especially number theory, is no more difficult than any other subject

If you have the willingness, patience, and time to sit down and do the work, then youwill find number theory worth studying and this book worth studying from; you willfind that number theory can be fun, and fun can be number theory Remember thatlearning mathematics is a step-by-step matter Do your work regularly and system-atically; review earlier chapters every week, since things must be fresh in your mind

to apply them and to build on them In this way, you will enjoy the subject and feelconfident to explore more I look forward to hearing from you with your commentsand suggestions In the meantime, enjoy the book

Thomas Koshy

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1 Fundamentals

Tell me and I will forget.

Show me and I will remember Involve me and I will understand.

— CONFUCIUS

he outstanding German mathematician Karl Friedrich Gauss (1777–1855)

Tonce said, “Mathematics is the queen of the sciences and arithmetic the queen

of mathematics.” “Arithmetic,” in the sense Gauss uses it, is number theory,which, along with geometry, is one of the two oldest branches of mathematics Num-ber theory, as a fundamental body of knowledge, has played a pivotal role in thedevelopment of mathematics And as we will see in the chapters ahead, the study ofnumber theory is elegant, beautiful, and delightful

A remarkable feature of number theory is that many of its results are within thereach of amateurs These results can be studied, understood, and appreciated with-out much mathematical sophistication Number theory provides a fertile ground forboth professionals and amateurs We can also find throughout number theory manyfascinating conjectures whose proofs have eluded some of the most brilliant mathe-maticians We find a great number of unsolved problems as well as many intriguingresults

Another interesting characteristic of number theory is that although many of itsresults can be stated in simple and elegant terms, their proofs are sometimes longand complicated

Generally speaking, we can define “number theory” as the study of the properties

of numbers, where by “numbers” we mean integers and, more specifically, positiveintegers

Studying number theory is a rewarding experience for several reasons First, ithas historic significance Second, integers, more specifically, positive integers, are

1

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A Greek Stamp

Honoring

Pythagoras

The Island of Samos

Pythagoras (ca 572–ca 500B.C ), a Greek pher and mathematician, was born on the Aegean island of Samos After extensive travel and studies, he returned home around 529 B.C only to find that Samos was under tyranny, so he migrated to the Greek port

philoso-of Crontona, now in southern Italy There he founded the famous Pythagorean school among the aristo- crats of the city Besides being an academy for philosophy, mathematics, and natural science, the school became the center of a closely knit brotherhood shar- ing arcane rites and observances The brotherhood ascribed all its discoveries to the master.

A philosopher, Pythagoras taught that number was the essence of everything, and

he associated numbers with mystical powers He also believed in the transmigration of the soul, an idea he might have borrowed from the Hindus.

Suspicions arose about the brotherhood, leading to the murder of most of its members The school was destroyed in a political uprising It is not known whether Pythagoras escaped death or was killed.

the building blocks of the real number system, so they merit special recognition.Third, the subject yields great beauty and offers both fun and excitement Finally,the many unsolved problems that have been daunting mathematicians for centuriesprovide unlimited opportunities to expand the frontiers of mathematical knowledge.Goldbach’s conjecture (Section 2.5) and the existence of odd perfect numbers (Sec-tion 8.3) are two cases in point Modern high-speed computers have become a pow-erful tool in proving or disproving such conjectures

Although number theory was originally studied for its own sake, today it hasintriguing applications to such diverse fields as computer science and cryptography(the art of creating and breaking codes)

The foundations for number theory as a discipline were laid out by the Greek

mathematician Pythagoras and his disciples (known as the Pythagoreans) The

Pythagorean brotherhood believed that “everything is number” and that the centralexplanation of the universe lies in number They also believed some numbers havemystical powers The Pythagoreans have been credited with the invention of am-icable numbers, perfect numbers, figurate numbers, and Pythagorean triples Theyclassified integers into odd and even integers, and into primes and composites.Another Greek mathematician, Euclid (ca 330–275B.C.), also made significantcontributions to number theory We will find many of his results in the chapters tofollow

We begin our study of number theory with a few fundamental properties of tegers

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in-Little is known about Euclid’s life He was on the faculty at the University of

Alexan-dria and founded the AlexanAlexan-drian School of Mathematics When the Egyptian ruler King Ptolemy I asked Euclid, the father of geometry, if there were an easier way to learn geometry than by studying The Elements, he replied, “There is no royal road

to geometry.”

The German mathematician Hermann Minkowski (1864–1909) once remarked, tegral numbers are the fountainhead of all mathematics.” We will come to appreciatehow important his statement is In fact, number theory is concerned solely with inte-

“In-gers The set of integers is denoted by the letter Z:†

Z= { , −3, −2, −1, 0, 1, 2, 3, }

Whenever it is convenient, we write “x ∈ S” to mean “x belongs to the set S”;

“x / ∈ S” means “x does not belong to S.” For example, 3 ∈ Z, but√3 /∈ Z.

We can represent integers geometrically on the number line, as in Figure 1.1.

Figure 1.1

The integers 1, 2, 3, are positive integers They are also called natural

num-bers or counting numnum-bers; they lie to the right of the origin on the number line We denote the set of positive integers by Z+or N:

Z+= N = {1, 2, 3, }

† The letter Z comes from the German word Zahlen for numbers.

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Leopold Kronecker (1823–1891) was born in 1823 into a well-to-do family in Liegnitz,

Prussia (now Poland) After being tutored privately at home during his early years and then attending a preparatory school, he went on to the local gymnasium, where he excelled in Greek, Latin, Hebrew, mathematics, and philosophy There he was fortu- nate to have the brilliant German mathematician Ernst Eduard Kummer (1810–1893)

as his teacher Recognizing Kronecker’s mathematical talents, Kummer encouraged him to pursue independent scientific work Kummer later became his professor at the universities of Breslau and Berlin.

In 1841, Kronecker entered the University of Berlin and also spent time at the University of Breslau He attended lectures by Dirichlet, Jacobi, Steiner, and Kummer Four years later he received his Ph.D in mathematics.

Kronecker’s academic life was interrupted for the next 10 years when he ran his uncle’s business less, he managed to correspond regularly with Kummer After becoming a member of the Berlin Academy of Sciences in 1861, Kronecker began his academic career at the University of Berlin, where he taught unpaid until 1883; he became a salaried professor when Kummer retired.

Nonethe-In 1891, his wife died in a fatal mountain climbing accident, and Kronecker, devastated by the loss, cumbed to bronchitis and died four months later.

suc-Kronecker was a great lover of the arts, literature, and music, and also made profound contributions to ber theory, the theory of equations, elliptic functions, algebra, and the theory of determinants The vertical bar notation for determinants is his creation.

num-The German mathematician Leopold Kronecker wrote, “God created the naturalnumbers and all else is the work of man.” The set of positive integers, together with 0,

forms the set of whole numbers W:

W= {0, 1, 2, 3, }

Negative integers, namely, , −3, −2, −1, lie to the left of the origin Notice

that 0 is neither positive nor negative

We can employ positive integers to compare integers, as the following definitionshows

The Order Relation

Let a and b be any two integers Then a is less than b, denoted by a < b, if there

exists a positive integer x such that a + x = b, that is, if b − a is a positive integer.

When a < b, we also say that b is greater than a, and we write b > a.†

† The symbols < and > were introduced in 1631 by the English mathematician Thomas Harriet

(1560–1621).

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If a is not less than b, we write a ≮ b; similarly, a ≯ b indicates a is not greater than b.

It follows from this definition that an integer a is positive if and only if a > 0 Given any two integers a and b, there are three possibilities: either a < b, a = b,

or a > b This is the law of trichotomy Geometrically, this means if a and b are any

two points on the number line, then either point a lies to the left of point b, the two points are the same, or point a lies to the right of point b.

We can combine the less than and equality relations to define the less than or

equal to relation If a < b or a = b, we write a ≤ b.†Similarly, a ≥ b means either

a > b or a = b Notice that a < b if and only if a ≥ b.

We will find the next result useful in Section 3.4 Its proof is fairly simple and is

an application of the law of trichotomy

THEOREM‡1.1 Let min{x, y} denote the minimum of the integers x and y, and max{x, y} their

maxi-mum Then min{x, y} + max{x, y} = x + y.§

PROOF (by cases) case 1 Let x ≤ y Then min{x, y} = x and max{x, y} = y, so min{x, y}+max{x, y} =

For example,|5| = 5, |−3| = −(−3) = 3, |π| = π, and |0| = 0.

Geometrically, the absolute value of a number indicates its distance from theorigin on the number line

Although we are interested only in properties of integers, we often need todeal with rational and real numbers also Floor and ceiling functions are two suchnumber-theoretic functions They have nice applications to discrete mathematics andcomputer science

† The symbols ≤ and ≥ were introduced in 1734 by the French mathematician P Bouguer.

‡ A theorem is a (major) result that can be proven from axioms or previously known results.

§ Theorem 1.1 is true even if x and y are real numbers.

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Floor and Ceiling Functions

The floor of a real number x, denoted by x, is the greatest integer ≤ x The ceiling

of x, denoted by †The floor of x rounds down x, whereas the ceiling of x rounds up Accordingly, if x / ∈ Z, the floor of x is the nearest integer to

the left of x on the number line, and the ceiling of x is the nearest integer to the right

of x, as Figure 1.2 shows The floor function f (x) = x and the ceiling function

The floor function comes in handy when real numbers are to be truncated or

rounded off to a desired number of decimal places For example, the real number π=

3.1415926535 truncated to three decimal places is given by 1000π/1000 = 3141/1000 = 3.141; on the other hand, π rounded to three decimal places is

1000π + 0.5/1000 = 3.142.

There is yet another simple application of the floor function Suppose we dividethe unit interval [0, 1) into 50 subintervals of equal length 0.02 and then seek to

determine the subinterval that contains the number 0.4567 Since0.4567/0.02 +

1= 23, it lies in the 23rd subinterval More generally, let 0 ≤ x < 1 Then x lies in

the subintervalx/0.02 + 1 = 50x + 1.

The following example presents an application of the ceiling function to day life

every-EXAMPLE 1.1 (The post-office function) In 2006, the postage rate in the United States for a

first-class letter of weight x, not more than one ounce, was 39¢; the rate for each additional

ounce or a fraction thereof up to 11 ounces was an additional 24¢ Thus, the postage

p(x) for a first-class letter can be defined as p(x) For instance, the postage for a letter weighing 7.8 ounces is p(7.8) = 0.39 +

† These two notations and the names, floor and ceiling, were introduced by Kenneth E Iverson in the

early 1960s Both notations are variations of the original greatest integer notation[x].

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Some properties of the floor and ceiling functions are listed in the next theorem.

We shall prove one of them; the others can be proved as routine exercises

1 The English mathematician Augustus DeMorgan,

who lived in the 19th century, once remarked that he

was x years old in the year x2 When was he born?

Evaluate each, where x is a real number.

7 There are four integers between 100 and 1000 that are

each equal to the sum of the cubes of its digits Three

of them are 153, 371, and 407 Find the fourth

num-ber (Source unknown.)

8 An n-digit positive integer N is a Kaprekar number

if the sum of the number formed by the last n digits

in N2, and the number formed by the first n (or n− 1)

digits in N2equals N For example, 297 is a Kaprekar

number since 2972= 88209 and 88 + 209 = 297

There are five Kaprekar numbers < 100 Find them.

9 Find the flaw in the following “proof”:

Let a and b be real numbers such that a = b Then

ab = b2

a2− ab = a2− b2

Factoring, a(a − b) = (a + b)(a − b) Canceling

a − b from both sides, a = a + b Since a = b,

this yields a = 2a Canceling a from both sides,

we get 1 = 2.

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D R Kaprekar (1905–1986) was born in Dahanu, India, near Bombay After losing his mother at the age of

eight, he built a close relationship with his astrologer-father, who passed on his knowledge to his son He tended Ferguson College in Pune, and then graduated from the University of Bombay in 1929 He was awarded the Wrangler R P Paranjpe prize in 1927 in recognition of his mathematical contributions A prolific writer in recreational number theory, he worked as a schoolteacher in Devlali, India, from 1930 until his retirement in 1962.

at-Kaprekar is best known for his 1946 discovery of the at-Kaprekar constant 6174 It took him about three

years to discover the number: Take a four-digit number a, not all digits being the same; letadenote the number

obtained by rearranging its digits in nondecreasing order anda denote the number obtained by rearranging its

digits in nonincreasing order Repeat these steps withb = a− a and its successors Within a maximum of eight steps, this process will terminate in 6174 It is the only integer with this property.

10 Express 635,318,657 as the sum of two fourth powers

in two different ways (It is the smallest number with

this property.)

11 The integer 1105 can be expressed as the sum of two

squares in four different ways Find them

12 There is exactly one integer between 2 and 2× 1014

that is a perfect square, a cube, and a fifth power Find

it (A J Friedland, 1970)

13 The five-digit number 2xy89 is the square of an

in-teger Find the two-digit number xy (Source:

Mathe-matics Teacher)

14 How many perfect squares can be displayed on a

15-digit calculator?

15 The number sequence 2, 3, 5, 6, 7, 10, 11, consists

of positive integers that are neither squares nor cubes

Find the 500th term of this sequence (Source:

The distance from x to y on the number line, denoted by

d(x, y), is defined by d(x, y) = |y − x| Prove each, where

x, y, and z are any integers.

33 Let max{x, y} denote the maximum of x and y, and

min{x, y} their minimum, where x and y are any

inte-gers Prove that max{x, y} − min{x, y} = |x − y|.

34 A round-robin tournament has n teams, and each team

plays at most once in a round Determine the

mini-mum number of rounds f (n) needed to complete the

tournament (Romanian Olympiad, 1978)

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Joseph Louis Lagrange (1736–1813), who ranks with Leonhard Euler as one of the

greatest mathematicians of the 18th century, was the eldest of eleven children in a wealthy family in Turin, Italy His father, an influential cabinet official, became bank- rupt due to unsuccessful financial speculations, which forced Lagrange to pursue a profession.

As a young man studying the classics at the College of Turin, his interest in ematics was kindled by an essay by astronomer Edmund Halley on the superiority of the analytical methods of calculus over geometry in the solution of optical problems.

math-In 1754 he began corresponding with several outstanding mathematicians in Europe The following year, Lagrange was appointed professor of mathematics at the Royal Artillery School in Turin Three years later, he helped to found a society that later became the Turin Academy of Sciences While at Turin, Lagrange developed revolu- tionary results in the calculus of variations, mechanics, sound, and probability, winning the prestigious Grand Prix

of the Paris Academy of Sciences in 1764 and 1766.

In 1766, when Euler left the Berlin Academy of Sciences, Frederick the Great wrote to Lagrange that “the greatest king in Europe” would like to have “the greatest mathematician of Europe” at his court Accepting the invitation, Lagrange moved to Berlin to head the Academy and remained there for 20 years When Frederick died

in 1786, Lagrange moved to Paris at the invitation of Louis XVI Lagrange was appointed professor at the École Normale and then at the École Polytechnique, where he taught until 1799.

Lagrange made significant contributions to analysis, analytical mechanics, calculus, probability, and number theory, as well as helping to set up the French metric system.

We will find both the summation and the product notations very useful throughoutthe remainder of this book First, we turn to the summation notation

The Summation Notation

Sums, such as a k + a k+1+ · · · + a m, can be written in a compact form using the

summation symbol

(the Greek uppercase letter sigma), which denotes the word sum The summation notation was introduced in 1772 by the French mathematician

Joseph Louis Lagrange

A typical term in the sum above can be denoted by a i, so the above sum is the

sum of the numbers a i as i runs from k to m and is denoted by

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The variable i is the summation index The values k and m are the lower and upper limits of the index i The “i=” above theis usually omitted:

The index i is a dummy variable; we can use any variable as the index without

affecting the value of the sum, so

The following results are extremely useful in evaluating finite sums They can

be proven using mathematical induction, presented in Section 1.3

THEOREM 1.3 Let n be any positive integer and c any real number, and a1, a2, , a n and b1,

b2, , b nany two number sequences Then

n

i=1

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(These results can be extended to any lower limit k ∈ Z.)

The following example illustrates this theorem

The summation notation can be extended to sequences with index sets I as their

domains For instance,

i ∈I a i denotes the sum of the values of a i as i runs over the

various values in I.

As an example, let I = {0, 1, 3, 5} Then

i ∈I ( 2i + 1) represents the sum of the values of 2i + 1 with i ∈ I, so

i ∈I ( 2i + 1) = (2 · 0 + 1) + (2 · 1 + 1) + (2 · 3 + 1) + (2 · 5 + 1) = 22

Often we need to evaluate sums of the form

P

a ij , where the subscripts i and j satisfy certain properties P (Such summations are used in Chapter 8.)

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For example, let I = {1, 2, 3, 4} Then

1≤i<j≤4 ( 2i + 3j) denotes the sum of the values of 2i + 3j, where 1 ≤ i < j ≤ 4 This can be abbreviated as

i<j ( 2i + 3j) pro-

vided the index set is obvious from the context To find this sum, we must consider

every possible pair (i, j), where i, j ∈ I and i < j Thus,

i<j ( 2i + 3j) = (2 · 1 + 3 · 2) + (2 · 1 + 3 · 3) + (2 · 1 + 3 · 4) + (2 · 2 + 3 · 3)

d = sum of positive integers d, where d is a factor of 6

= sum of positive factors of 6

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We now turn to the product notation.

The Product Notation

Again, i is just a dummy variable.

The following three examples illustrate this notation

The factorial function, which often arises in number theory, can be defined

using the product symbol, as the following example shows

EXAMPLE 1.6 The factorial function f (n) = n! (read n factorial) is defined by n! = n(n−1) · · · 2·1,

where 0! = 1 Using the product notation, f (n) = n! = n

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