Topics in the Theory of Numbers

302 Pages·2003·7.18 MB·English

by  Paul Erdős

Undergraduate Texts in Mathematics
Editors
s. Axler
F.W. Gehring
K.A. Ribet
Undergraduate Texts in Mathematics
Abbott: Understanding Analysis. Chambert-Loir: A Field Guide to Algebra
Anglin: Mathematics: A Concise History Childs: A Concrete Introduction to
and Philosophy. Higher Algebra. Second edition.
Readings in Mathematics. ChungtA itSahlia: Elementary Probability
Anglin/Lambek: The Heritage of Theory: With Stochastic Processes and
Thales. an Introduction to Mathematical
Readings in Mathematics. Finance. Fourth edition.
Apostol: Introduction to AnaIytic Cox/Little/O'Shea: Ideals, Varieties,
Number Theory. Second edition. and Algorithms. Second edition.
Armstrong: Basic Topology. Croom: Basic Concepts of Algebraic
Armstrong: Groups and Symmetry. Topology.
Ader: Linear Algebra Done Right. Curtis: Linear Algebra: An Introductory
Second edition. Approach. Fourth edition.
Beardon: Limits: A New Approach to Daepp/Gorkin: Reading, Writing, and
Real Analysis. Proving: A Closer Look at
BakINewman: Complex Analysis. Mathematics.
Second edition. Devlin: The Joy of Sets: Fundamentals
BanchofflWermer: Linear Algebra of Contemporary Set Theory.
Through Geometry. Second edition. Second edition.
Berberian: A First Course in Real Dixmier: General Topology.
Analysis. Driver: Why Math?
Bix: Conics and Cubics: A EbbinghauslFlumffhomas:
Concrete Introduction to Algebraic Mathematical Logic. Second edition.
Curves. Edgar: Measure, Topology, and FractaI
Br~maud: An Introduction to Geometry.
Probabilistic Modeling. Elaydi: An Introduction to Difference
Bressoud: Factorization and Primality Equations. Second edition.
Testing. Erd6s/Suranyi: Topics in the Theory of
Bressoud: Second Year Calculus. Numbers.
Readings in Mathematics. Estep: Practical Analysis in One Variable.
Brickman: Mathematical Introduction Exner: An Accompaniment to Higher
to Linear Programming and Game Mathematics.
Theory. Exner: Inside Calculus.
Browder: Mathematical Analysis: Fine/Rosenberger: The Fundamental
An Introduction. Theory of Algebra.
Buchmann: Introduction to Fischer: Intermediate Real Analysis.
Cryptography. Flanigan/Kazdan: Calculus Two: Linear
Buskes/van Rooij: Topological Spaces: and Nonlinear Functions. Second
From Distance to Neighborhood. edition.
Callahan: The Geometry of Spacetime: Fleming: Functions ofSeveral Variables.
An Introduction to Special and General Second edition.
Relavitity. Foulds: Combinatorial Optimization for
Carter/van Brunt: The Lebesgue Undergraduates.
Stieltjes Integral: A Practical Foulds: Optimization Techniques: An
Introduction. Introduction.
Cederberg: A Course in Modem Franklin: Methods ofMathematical
Geometries. Second edition. Economics.
(continued after index)
Paul Erd6s
Jânos Surânyi
Topics in the Theory
of Numbers
Translated by Barry Guiduli
With 32 Illustrations
~
Springer
Paul Erdos Jânos Surânyi
(deceased) Department of Algebra and
Number Theory
Eătvăs Lorând University
Pâzmâny Peter Setany llC
Budapest, H-1l17
Hungary
suranyi@;s.elte.hu
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East Hall University of California,
University University of Michigan Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840
USA USA USA
[email protected] [email protected] [email protected]
Mathematics Subject Classification (2000): 51-01
Library of Congress Cataloging-in-Publication Data
Erdos, Paul, 1913-
[Vâlogatott fejezetek a szâmelmeletbol. English]
Topics in the theory of numbers / Paul Erdos, Jânos Surânyi.
p. cm. - (Undergraduate texts in mathematics)
Includes bibliographica1 references and index.
ISBN 978-1-4612-6545-0 ISBN 978-1-4613-0015-1 (eBook)
DOI 10.1007/978-1-4613-0015-1
1. Number theory. 1. Surânyi Jânos, 1918- II. Title. III. Series.
QA241 .E7613 2002
512'.7-dc21 2002067532
ISBN 978-1-4612-6545-0 Printed on acid-free paper.
© 2003 Springer Science+Business Media New York
Originally published by Springer Science+Business Media, Inc. in 2003
Softcover reprint ofthe hardcover 1st edition 2003
AlI rights reserved. This work may not be translated or copied in whole or in part with
out the written permission ofthe publisher Springer Science+Business Media,LLC
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Preface to the Second Edition!
Thefirst editionofour bookhaslongsincebeenout ofprint, andmanypeople
who regard this book as the source of their first mathematical inspiration,
and others whose only contact with the book was in a library (the book has
disappeared from many),havebeen pushing foralongtimefor a newedition.
We happily comply with all these urgings. We would especially like to thank
MIKLOS SIMONOVITS, who not only encouraged our work unwaveringly, but
also gave precious help.
The goals of the book as expressed in the preface to the first edition
are unchanged. Some sections, however, have undergone significant changes.
A new Chapter 2 was written about congruences. The several important
theoremsappearingin thisnewchapteroriginallyappeared inthefirst edition
stated in terms of divisibility. However, we felt it useful to introduce the
notion of congruence. We have included a more thorough discussion of the
subject, including second-degree congruences. Here as well we have chosen
an interesting, albeit lesser known, path to follow.
In the nearly 40 long years that have passed since the first edition, many
important changes have occurred in all of mathematics, as well as in num
ber theory in particular, and to a certain degree the authors' interests have
changed as well. These changes have resulted in two important changes in
the book. We have omitted a proof of a number-theoretic theorem about
polynomials, as well as the accompanying section on polynomial arithmetic.
Furthermore, the section concerning the number of elements in a series no
threeofwhoseelements formsanarithmeticprogressionhasbeensignificantly
shortened; since the first edition, SZEMEREDI has established a very deep re
sult for these series. On the other hand, we have included a few new results
on so-called Sidon sets, sets ofnumbers that do not contain the differenceof
any twoelements. (See Section 6.21 and subsequent sections.)
We remarked in the preface to the first edition that one charm of the
integers is that easily stated problems, which often sound simple, are often
very difficult and sometimes even hopeless given the state of our current
knowledge. For instance, in a 1912 lecture at an international mathematical
congress, EDMUND LANDAUmentioned four old conjectures that appeared
hopeless at that time:
1 This is an abridged translation ofthe original Hungarian text.
vi Preface to the Second Edition
• Every even number greater than two is the sum of two primes.
• Between any consecutive squares there is a prime number.
• There are infinitely many twin primes (these are primes that are consecu
tive odd integers).
• There are infinitely many primes of the form n2 +1.
Today, his statement could be reiterated. During the more than 80 years
that have passed, much intensive research has been conducted on all of these
conjectures, and we now know that every large enough odd number is the
sum of three primes, and every even number can be written as the sum of a
prime and a number that is divisible by at most two different primes; there
are infinitely many consecutive odd integers for which one is a prime and the
other has at most two prime divisors; for infinitely many n, there is a prime
betweenn2 and n2+nl.1; forinfinitelymanyn, n2+1 has at most two distinct
prime divisors. Unfortunately, the methods used to achieve these rather deep
results cannotbe generalizedtoprovethemoregeneralconjectures, andgiven
the state of mathematics today, the resolution of these conjectures can still
be called hopeless.
As we mentioned, we restricted ourselves to results that can be estab
lished by using only elementary tools. This does not mean, however, that
their proofs are simple. Juxtaposed with rather simple ideas, some rather
deep proofs occur, especially from Chapter 5onward. Amongthese, the most
difficult is Theorem 11 in Chapter 5 (Sections 20-22.), Theorems 4 and 5
in Chapter 8 (Sections 11*-13* and 16*-18*, respectively). We have marked
these sections with stars; for less experienced readers, we recommend skim
ming these sections during the first reading.
We express our deepest gratitude to IMREZ. RUZSA for reading through
the entire manuscript and to IMREBARANY for reading through Chapter 4;
they greatly helpedourwork. Itwould be verydifficult to list all those people
whose comments and suggestions from the first edition have helped improve
this edition. Here wecollectively express our appreciation to one and all. For
selfless help with the revisions ofthe book, weexpress our deepest thanks to
ANTAL BALOG, ROBERT FREUD,KALMAN GYORY, JANOS PINTZ,ANDRAS
SARKOZY, and VERA T. Sos and additionally to all those people who sup
ported us with their comments and advice.
Budapest, August 1995 Paul Erdos, Janos Suranyi
Preface to the First Edition?
The numbers weknow best are the integers, but these are perhaps the most
elusive as well. Number theory, the branch ofmathematics that studies their
properties,isarepositoryofinterestingand quitevaried problems,sometimes
impossiblydifficult. We have gathered together a collection ofproblems from
various topics in this field of mathematics that we find beautiful, intrigu
ing, and from a certain point of view instructive. We hope that others take
pleasure in them as well.
In addition to reveling in the beauty ofthe problems themselves,wehave
tried to give glimpses into the deeper related mathematics. We endeavored
to show the living mathematics, giving examples of problems that can be
solved using elementary tools, and which often have related problems whose
solutions require very difficult lemmas and deep ideas; in fact, among these
related problems weoften come across ones whose solutions seem hopeless in
light ofthe present state of knowledge.
In our book we present only problems whose solutions can be obtained
using elementary methods. Wedo not assumeany priorknowledge ofnumber
theory.
Wetried to useonly a fewresults from other areasofmathematics, which
we were obliged to use without proof because their proofs fall outside the
scope of this book, and by trying to include their proofs, the book would
have grown too big. Among these, we list the most important ones. The
reader can find their proofs in any elementary textbook, but the discussions
within this book can be followedwithout any difficulty if the reader accepts
the results without proof.
Formany oftheproofs wetryto provide motivationasto whyweapproach
problems in the given manner, and we try to present the important lines of
thought needed to arriveat the solution. In theseinstances, wehave borrowed
wisdom from GYORGY POLYA'S books, and wewould liketo thank professors
ROZSA PETER and LASZLO KALMAR. (It is another question as to how well
wehave succeeded.)
We have included exercises after the different problem topics. In some
instances these are easy problems whose solutions build upon or are based
on the established results.Theother instances the problems give results that
2 This is an abridged translation of the original Hungarian text.
viii Prefaceto the First Edition
extendthethemesdiscussed and are ofvarying degrees ofdifficulty.Themost
difficult ofthese wehave indicated by a star. Wegivehints to their solutions
in the appendix of this book. When the problems are due to other authors,
wehave indicated their sources, in so much as it was possible to determine.
We have divided the chapters into sections. In the interest ofreadability,
however, we have not given these names. The reader will find the detailed
content in the table of contents, with the sections grouped by subject. The
sections, theorems, formulas,and footnotes are numberedin increasingorder,
starting anew in each chapter. Werefer to these using only the number when
they are in the same chapter, and using the number prefixed by the chapter
number when they are in other chapters.
The chapters of the book build little upon each other, except for the
fundamental notions given in the first chapter, and can essentially be read
independently ofeach other after the first chapter, with only the material of
the third and fourth chapters [with the new numbering] being related.
PAL TURAN strongly supported our work, starting from the selection of
subjectmatter, and furthermorewereceivedmany interestingcommentsfrom
him and ROZSA PETERupon reading our entire manuscript.Withgratitude,
wethank them both.
Budapest, July 1959 Paul Erdos, Janos Suranyi
Preface to the English Translation
The death of Professor PAUL ERDOS on September 20, 1996, in Warsaw,
was a great loss to the world mathematical community. I can attest to the
fact that until the very end he kept a watchful eye on this translation and
proposed several new results that arose since the writing ofthe second Hun
garianedition. Theevening before he left forWarsaw weweretogether in my
apartment discussing the translation.
For the translation ofthe manuscript weexpress our deepest gratitude to
the translator, BARRY GUIDULI, who not only translated the text but also
provided many interesting comments and suggestions. Best thanks are also
due to ROMY VARGA, who assisted him greatly in the translation.
We also thank ZOLTAN KIRALY for indispensable technical help. To the
Alfred Renyi Mathematical Research Institute ofthe Hungarian Academy of
Science, as well as to the Computer Science and Mathematics Departments
of Eotvos Lorand University, we are indebted for their continuous support.
Without their gracious help and support, this project would not have been
realized.
Budapest, April 2000 Janos Suranyi