Graduate Texts in Mathematics S Axler Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo 195 Editorial Board F.W Gehring K.A Ribet Graduate Texts in Mathematics TAKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OxTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 ALEXANDER/WERMER Several Complex Variables and Banach Algebras 3rd ed HILTON/STAMMBACH A Course in 36 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 1.2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GOLUBITSKY/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HuSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol.1 ZARISKI/SAMUEL Commutative Algebra Vol.11 JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory KELLEY/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 37 38 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 KEMENY/SNELL/KNAPP Denumerable Markov Chains 2nd ed APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEVE Probability Theory I 4th ed LOEVE Probability Theory II 4th ed MoiSE Geometric Topology in Dimensions and SACHS/WU General Relativity for Mathematicians GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry MANIN A Course in Mathematical Logic GRAVER/WATKINS Combinatories with Emphasis on the Theory of Graphs BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis MASSEY Algebraic Topology: An Introduction CROWELL/FOX Introduction to Knot Theory KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed LANG Cyclotomic Fields ARNOLD Mathematical Methods in Classical Mechanics 2nd ed WHITEHEAD Elements of Homotopy Theory (continued after index) Melvyn B Nathanson Elementary Methods in Number Theory Springer Melvyn B Nathanson Department of Mathematics Lehman College (CUNY) Bronx, NY 10468 USA nathansn@alpha.lehman.cuny.edu Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring K.A Ribet Mathematics Department East Hall University of Michigan Ann Arbor, Ml 48109 USA Mathematics Department University of California Berkeley, CA 94720-3840 USA Mathematics Subject Classification (1991): 11-01 Library of Congress Cataloging-in-Publication Data Nathanson, Melvyn B (Melvyn Bernard), 1944Elementary methods in number theory / Melvyn B Nathanson p cm.—(Graduate texts in mathematics; 195) Includes bibliographical references and index ISBN 0-387-98912-9 (hardcover: alk paper) Number theory I Title II Series QA241.N3475 2000 512'.7—dc21 99-42812 ©2000 Melvyn B Nathanson 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 epecially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may be accordingly used freely by anyone ISBN 0-387-98912-9 Springer-Verlag New York Berlin Heidelberg SPIN 10742484 To Paul Erd˝os, 1913–1996, a friend and collaborator for 25 years, and a master of elementary methods in number theory Preface Arithmetic is where numbers run across your mind looking for the answer Arithmetic is like numbers spinning in your head faster and faster until you blow up with the answer KABOOM!!! Then you sit back down and begin the next problem Alexander Nathanson [99] This book, Elementary Methods in Number Theory, is divided into three parts Part I, “A first course in number theory,” is a basic introduction to elementary number theory for undergraduate and graduate students with no previous knowledge of the subject The only prerequisites are a little calculus and algebra, and the imagination and perseverance to follow a mathematical argument The main topics are divisibility and congruences We prove Gauss’s law of quadratic reciprocity, and we determine the moduli for which primitive roots exist There is an introduction to Fourier analysis on finite abelian groups, with applications to Gauss sums A chapter is devoted to the abc conjecture, a simply stated but profound assertion about the relationship between the additive and multiplicative properties of integers that is a major unsolved problem in number theory The “first course” contains all of the results in number theory that are needed to understand the author’s graduate texts, Additive Number Theory: The Classical Bases [104] and Additive Number Theory: Inverse Problems and the Geometry of Sumsets [103] viii Preface The second and third parts of this book are more difficult than the “first course,” and require an undergraduate course in advanced calculus or real analysis Part II is concerned with prime numbers, divisors, and other topics in multiplicative number theory After deriving properties of the basic arithmetic functions, we obtain important results about divisor functions, and we prove the classical theorems of Chebyshev and Mertens on the distribution of prime numbers Finally, we give elementary proofs of two of the most famous results in mathematics, the prime number theorem, which states that the number of primes up to x is asymptotically equal to x/ log x, and Dirichlet’s theorem on the infinitude of primes in arithmetic progressions Part III, “Three problems in additive number theory,” is an introduction to some classical problems about the additive structure of the integers The first additive problem is Waring’s problem, the statement that, for every integer k ≥ 2, every nonnegative integer can be represented as the sum of a bounded number of kth powers More generally, let f (x) = ak xk + ak−1 xk−1 + · · · + a0 be an integer-valued polynomial with ak > such that the integers in the set A(f ) = {f (x) : x = 0, 1, 2, } have no common divisor greater than one Waring’s problem for polynomials states that every sufficiently large integer can be represented as the sum of a bounded number of elements of A(f ) The second additive problem is sums of squares For every s ≥ we denote by Rs (n) the number of representations of the integer n as a sum of s squares, that is, the number of solutions of the equation n = x21 + · · · + x2s in integers x1 , , xs The shape of the function Rs (n) depends on the parity of s In this book we derive formulae for Rs (n) for certain even values of s, in particular, for s = 2, 4, 6, 8, and 10 The third additive problem is the asymptotics of partition functions A partition of a positive integer n is a representation of n in the form n = a1 + · · · + ak , where the parts a1 , , ak are positive integers and a1 ≥ · · · ≥ ak The partition function p(n) counts the number of partitions of n More generally, if A is any nonempty set of positive integers, the partition function pA (n) counts the number of partitions of n with parts belonging to the set A We shall determine the asymptotic growth of p(n) and, more generally, of pA (n) for any set A of integers of positive density This book contains many examples and exercises By design, some of the exercises require old-fashioned manipulations and computations with pencil and paper A few exercises require a calculator Number theory, after all, begins with the positive integers, and students should get to know and love them This book is also an introduction to the subject of “elementary methods in analytic number theory.” The theorems in this book are simple statements about integers, but the standard proofs require contour integration, Preface ix modular functions, estimates of exponential sums, and other tools of complex analysis This is not unfair In mathematics, when we want to prove a theorem, we may use any method The rule is “no holds barred.” It is OK to use complex variables, algebraic geometry, cohomology theory, and the kitchen sink to obtain a proof But once a theorem is proved, once we know that it is true, particularly if it is a simply stated and easily understood fact about the natural numbers, then we may want to find another proof, one that uses only “elementary arguments” from number theory Elementary proofs are not better than other proofs, nor are they necessarily easy Indeed, they are often technically difficult, but they satisfy the aesthetic boundary condition that they use only arithmetic arguments This book contains elementary proofs of some deep results in number theory We give the Erd˝os-Selberg proof of the prime number theorem, Linnik’s solution of Waring’s problem, Liouville’s still mysterious method to obtain explicit formulae for the number of representations of an integer as the sum of an even number of squares, and Erd˝os’s method to obtain asymptotic estimates for partition functions Some of these proofs have not previously appeared in a text Indeed, many results in this book are new Number theory is an ancient subject, but we still cannot answer the simplest and most natural questions about the integers Important, easily stated, but still unsolved problems appear throughout the book You should think about them and try to solve them Melvyn B Nathanson1 Maplewood, New Jersey November 1, 1999 Supported in part by grants from the PSC-CUNY Research Award Program and the NSA Mathematical Sciences Program This book was completed while I was visiting the Institute for Advanced Study in Princeton, and I thank the Institute for its hospitality I also thank Jacob Sturm for many helpful discussions about parts of this book Notation and Conventions We denote the set of positive integers (also called the natural numbers) by N and the set of nonnegative integers by N0 The integer, rational, real, and complex numbers are denoted by Z, Q, R, and C, respectively The absolute value of z ∈ C is |z| We denote by Zn the group of lattice points in the n-dimensional Euclidean space Rn The integer part of the real number x, denoted by [x], is the largest integer that is less than or equal to x The fractional part of x is denoted by {x} Then x = [x] + {x}, where [x] ∈ Z, {x} ∈ R, and ≤ {x} < In computer science, the integer part of x is often called the floor of x, and denoted by x The smallest integer that is greater than or equal to x is called the ceiling of x and denoted by x We adopt the standard convention that an empty sum of numbers is equal to and an empty product is equal to Similarly, an empty union of subsets of a set X is equal to the empty set, and an empty intersection is equal to X We denote the cardinality of the set X by |X| The largest element in a finite set of numbers is denoted by max(X) and the smallest is denoted by min(X) Let a and d be integers We write d|a if d divides a, that is, if there exists an integer q such that a = dq The integers a and b are called congruent modulo m, denoted by a ≡ b (mod m), if m divides a − b A prime number is an integer p > whose only divisors are and p The set of prime numbers is denoted by P, and pk is the kth prime Thus, p1 = 2, p2 = 3, , p11 = 31, Let p be a prime number We write pr n References 501 [56] G H Hardy and J E Littlewood Some problems of Partitio Numerorum. 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Modern Number Theory, volume 84 of Graduate Texts in Mathematics SpringerVerlag, New York, 2nd edition, 1990 [73] H Iwaniec Almost-primes represented by quadratic polynomials Inventiones Math., 47:171–188, 1978 [74] H Iwaniec Topics in Classical Automorphic Forms, volume 17 of Graduate Studies in Mathematics Amer Math Soc., Providence, 1997 [75] S M Johnson On the representations of an integer as a sum of products Trans Amer Math Soc., 76:177–189, 1954 [76] E Kamke Verallgemeinerung des Waring-Hilbertschen Satzes Math Annalen, 83:85112, 1921 ă [77] J Karamata Uber die Hardy–Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes Math Zeit., 32:319–320, 1930 [78] A Ya Khinchin Three Pearls of Number Theory Dover Publications, Mineola, NY, 1998 This translation from the Russian of the second (1948), revised edition was published originally by Graylock Press in 1952 [79] M Kneser Abschăatzungen der asymptotischen Dichte von Summenmengen Math Z., 58:459–484, 1953 [80] C Knessl and J B Keller Partition asymptotics for recursion equations SIAM J Applied Math., 50:323–338, 1990 [81] M I Knopp Modular Functions in Analytic Number Theory Markham Publishing Co., Chicago, 1970 Reprinted by Chelsea Publishing Company in 1993 [82] Chao Ko On the diophantine equation x2 = y n + 1, xy = Scientia Sinica, 14:457–460, 1964 [83] N Koblitz A Course in Number Theory and Cryptography, volume 114 of Graduate Texts in Mathematics Springer-Verlag, New York, 2nd edition, 1994 [84] E E Kohlbecker Weak asymptotic properties of partitions Trans Amer Math Soc., 88:346–375, 1958 References 503 [85] R Kumanduri and C Romero Number Theory with Computer Applications Prentice Hall, Upper Saddle River, New Jersey, 1998 [86] A V Kuzel’ Elementary solution of Waring’s problem for polynomials by the method of Yu B Linnik Uspekhi Mat Nauk, 11:165–168, 1956 [87] E Landau Elementary Number Theory Chelsea Publishing Company, New York, 1966 [88] S Lang Old and new conjectured diophantine inequalities Bull Amer Math Soc., 23:37–75, 1990 [89] S Lang Algebra Addison-Wesley, Reading, Mass., 3rd edition, 1993 [90] S Lang Algebraic Number Theory, volume 110 of Graduate Texts in Mathematics Springer-Verlag, New York, 2nd edition, 1994 [91] V A Lebesgue Sur l’impossibilit´e, en nombres entiers, de l’´equation xm = y + Nouv Ann Math (1), 9:178–181, 1850 [92] W J LeVeque Reviews in Number Theory Amer Math Soc., Providence, 1974 [93] Yu V Linnik An elementary solution of Waring’s problem by Shnirel’man’s method Mat Sbornik NS, 12 (54):225–230, 1943 [94] J E Littlewood Sur la distribution des nombres premiers C R Acad Sci Paris, S´er A, 158:1869–1872, 1914 [95] Yu I Manin Classical computing, quantum computing, and Shor’s factorization algorithm In S´eminaire Bourbaki, 51`eme ann´ee, 1998– 99, pages 862–1—862–30 UFR de Math´ematiques de l’Universit´e Paris VII — Denis Diderot, Paris, 1999 [96] Yu I Manin and A A Panchishkin Number Theory I Introduction to Number Theory, volume 49 of Encyclopedia of Mathematical Sciences Springer-Verlag, Berlin, 1995 [97] R C Mason Diophantine Equations over Function Fields, volume 96 of London Mathematical Society Lecture Notes Series Cambridge University Press, Cambridge, 1984 [98] M R Murty Artin’s conjecture for primitive roots Math Intelligencer, 10(4):59–67, 1988 [99] A P Nathanson “Arithmetic” Poem written in D’Ann Ippolito’s third grade class at Far Brook School, 1998 504 References [100] M B Nathanson An exponential congruence of Mahler Amer Math Monthly, 79:55–57, 1972 [101] M B Nathanson Sums of finite sets of integers Amer Math Monthly, 79:1010–1012, 1972 [102] M B Nathanson Catalan’s equation in K(t) Amer Math Monthly, 81:371–373, 1974 [103] M B Nathanson Additive Number Theory: Inverse Problems and the Geometry of Sumsets, volume 165 of Graduate Texts in Mathematics Springer-Verlag, New York, 1996 [104] M B Nathanson Additive Number Theory: The Classical Bases, volume 164 of Graduate Texts in Mathematics Springer-Verlag, New York, 1996 [105] M B Nathanson On Erd˝os’s elementary method in the asymptotic theory of partitions Preprint, 1998 [106] M B Nathanson Asymptotic density and the asymptotics of partition functions Acta Math Hungar., 87(1–2), 2000 [107] M B Nathanson Partitions with parts in a finite set Proc Amer Math Soc., 2000 To appear [108] M B Nathanson Additive Number Theory: Addition Theorems and the Growth of Sumsets In preparation, 2001 [109] V I Nechaev Waring’s Problem for Polynomials Izdat Akad Nauk SSSR, Moscow, 1951 [110] O Neugebauer The Exact Sciences in Antiquity Brown Univ Press, Providence, 2nd edition, 1957 Reprinted by Dover Publications in 1969 [111] J Neukirch 1999 Algebraic Number Theory Springer-Verlag, Berlin, [112] D J Newman Simple analytic proof of the prime number theorem Amer Math Monthly, 87:693–696, 1980 [113] A Nitaj La conjecture abc Enseignement Math., 42:3–24, 1996 [114] J Oesterl´e Nouvelles approches du “Th´eor`eme” de Fermat In S´eminaire Bourbaki, Volume 1987/88, Expos´es 686–699, volume 161– 162 of Ast´erisque Soci´et´e Math´ematique de France, Paris, 1988 [115] A G Postnikov A remark on an article by A G Postnikov and N P Romanov Uspehki Mat Nauk, 24(5(149)):263, 1969 References 505 [116] A G Postnikov and N P Romanov A simplification of A Selberg’s elementary proof of the asymptotic law of distribution of prime numbers Uspehki Mat Nauk (N.S.), 10(4(66)):75–87, 1955 [117] H Rademacher A convergent series for the partition function p(n) Proc Nat Acad Sci., 23:78–84, 1937 [118] H Rademacher On the partition function p(n) Proc London Math Soc., 43:241–254, 1937 [119] H Rademacher Topics in Analytic Number Theory Verlag, New York, 1973 Springer- [120] D Ramakrishnan and R J Valenza Fourier Analysis on Number Fields, volume 186 of Graduate Texts in Mathematics SpringerVerlag, New York, 1999 [121] S Ramanujan Some formulæ in the analytic theory of numbers Messenger of Mathemtics, 45:81–84, 1916 [122] G J Rieger Zu Linniks Lăosung des Waringschen Problems: Abschăatzung von g(n) Math Zeit., 60:213–239, 1954 [123] R L Rivest, A Shamir, and L M Adleman A method for obtaining digital signatures and public-key cryptosystems Communications of the ACM, 21:120–126, 1978 [124] A Schinzel Remarks on the paper “Sur certaines hypoth`eses concernant les nombres premiers” Acta Arith., 7:1–8, 1961/62 [125] A Schinzel and W Sierpi´ nski Sur certaines hypoth`eses concernant les nombres premiers Acta Arith., 4:185208, 1958 Erratum (1959), 259 ă [126] I Schur Uber die Gaußschen Summen Nachrichten k Gesell Gă ottingen, Math.-Phys Klasse, pages 147153, 1921 Reprinted in Gesammelte Abhandlungen, Band II, Springer-Verlag, Berlin, 1973 [127] A Selberg An elementary proof of Dirichlet’s theorem about primes in an arithmetic progression Annals Math., 50:297–304, 1949 In Collected Papers, volume I, pages 371–378, Springer-Verlag, Berlin, 1989 [128] A Selberg An elementary proof of the prime-number theorem Annals Math., 50:305–313, 1949 In Collected Papers, volume I, pages 379–387, Springer-Verlag, Berlin, 1989 [129] A Selberg An elementary proof of the prime-number theorem for arithmetic progressions Canadian J Math., 2:66–78, 1950 In Collected Papers, volume I, pages 398–410, Springer-Verlag, Berlin, 1989 506 References [130] A Selberg Reflections around the Ramanujan centenary In Collected Papers, volume I, pages 695–706 Springer-Verlag, Berlin, 1989 [131] J.-P Serre Cours d’Arithm´etique Presses Universitaires de France, Paris, 1970 [132] J.-P Serre A Course in Arithmetic, volume of Graduate Texts in Mathematics Springer-Verlag, New York, 1973 [133] P Shor Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer SIAM J Comput., 26:1484–1509, 1997 [134] J H Silverman Wieferich’s criterion and the abc conjecture J Number Theory, 30:226–237, 1988 [135] S Singh The Code Book: The Evolution of Secrecy from Mary, Queen of Scots to Quantum Cryptography Doubleday, New York, 1999 [136] E G Straus The elementary proof of the Prime Number Theorem Undated, unpublished manuscript [137] G Szekeres An asymptotic formula in the theory of partitions Quarterly J Math Oxford, 2:85–108, 1951 [138] G Szekeres Some asymptotic formulae in the theory of partitions (II) Quarterly J Math Oxford, 4:96–111, 1953 [139] R Taylor and A Wiles Ring-theoretic properties of certain Hecke algebras Annals Math., 141:533–572, 1995 [140] G Tenenbaum and M Mend`es-France The Prime Numbers and Their Distribution Amer Math Soc., Providence, 1999 [141] A Terras Fourier Analysis on Finite Groups and Applications Number 43 in London Mathematical Society Student Texts Cambridge University Press, Cambridge, 1999 [142] E C Titchmarsh The Theory of Functions Oxford University Press, Oxford, 2nd edition, 1939 [143] P Tur´an On a theorem of Hardy and Ramanujan J London Math Soc., 9:274–276, 1934 [144] P Tur´an On a New Method of Analysis and its Applications WileyInterscience, New York, 1984 [145] J V Uspensky and M A Heaslet Elementary Number Theory McGraw-Hill, New York, 1939 References 507 [146] Ya V Uspensky Asymptotic expressions of numerical functions occurring in problems concerning the partition of numbers into summands Bull Acad Sci de Russie, 14(6):199–218, 1920 [147] B L van der Waerden Science Awakening Science Editions, John Wiley & Sons, New York, 2nd edition, 1963 [148] R C Vaughan The Hardy–Littlewood Method Cambridge University Press, Cambridge, 2nd edition, 1997 [149] B A Venkov Elementary Number Theory Wolters-Noordhof Publishing, Groningen, the Netherlands, 1970 [150] I M Vinogradov On Waring’s theorem Izv Akad Nauk SSSR, Otd Fiz.-Mat Nauk, (4):393–400, 1928 English translation in Selected Works, pages 101–106, Springer-Verlag, Berlin, 1985 [151] S S Wagstaff Solution of Nathanson’s exponential congruence Math Comp., 33:1097–1100, 1979 [152] A Weil Number Theory for Beginners Springer-Verlag, New York, 1979 [153] A Weil Number Theory: An Approach through History From Hammurapi to Legendre Birkhăauser, Boston, 1984 [154] A Weil Basic Number Theory Classics in Mathematics SpringerVerlag, Berlin, 1995 Reprint of the 3rd edition, published in 1974 [155] A Wieferich Zum letzten Fermat’schen Satz J reine angew Math., 136:293–302, 1909 [156] A Wiles Modular elliptic curves and Fermat’s last theorem Annals Math., 141:443–531, 1995 [157] B M Wilson Proofs of some formulæ enunciated by Ramanujan Proc London Math Soc., 21:235–255, 1922 [158] Y Yang Inverse problems for partition functions Preprint, 1998 [159] D Zagier Newman’s short proof of the prime number theorem Amer Math Monthly, 104:705–708, 1997 Index abc conjecture, 185 abelian group, 10 abelian theorem, 486 abundant number, 241, 260 k-abundant, 260 primitive, 260 additive basis, 359 additive character, 325 additive set function, 133 algebraically closed field, 177 aliquot sequence, 243 arithmetic function, 57, 201 asymptotic basis, 359 asymptotic density, 244, 257, 360, 475 lower, 256, 482 upper, 256, 482 asymptotically stable basis, 360 basis, 359 asymptotic, 359 asymptotically stable, 360 of finite order, 359 of order h, 359 stable, 359 binary operation, 10 binary quadratic form, 108, 405 binomial coefficient, 8, 268 binomial polynomial, 357 Carmichael number, 76 Catalan conjecture, 186 Catalan equation, 184, 186 Catalan–Dickson problem, 244 Cauchy-Schwarz inequality, 139 ceiling function, xi character, 126 additive character, 325 complex character, 326 Dirichlet character, 326 even character, 326 induced, 328 multiplicative character, 326 odd character, 326 primitive, 328 principal character, 326 real character, 326 character group, 127 character table, 131 Chebyshev functions, 267 510 Index Chebyshev’s theorem, 271 ciphertext, 76 classical Gauss sum, 153 cofinite, 476 common divisor, 12 common multiple, 28 commutative group, 10 commutative ring, 48 comparative prime number theory, 351 complete set of residues, 46 completely additive, 27 completely multiplicative, 226 complex character, 326 composite number, 25 congruence abc conjecture, 191 congruence class, 46 congruent, 45 congruent polynomials, 90 conjugate divisor, 25, 405 continued fraction, 19 convergent, 23 convolution, 139 coset, 69 counting function, 256, 359, 475 cryptanalysis, 77 cryptography, 76 cusp form, 453 cyclic group, 70 deficient number, 241 degree of polynomial, 84 density, 256, 475 asymptotic, 360 Shnirel’man, 359 derivation, 175, 203 derivative, 116 diagonalizable operator, 146 difference operator, 357 difference set, 361 diophantine equation, 37 direct product of groups, 124 direct sum, 121 Dirichlet L-function, 330 Dirichlet character, 325, 326 Dirichlet convolution, 201 Dirichlet polynomial, 337 Dirichlet series, 337 Dirichlet’s divisor problem, 233 Dirichlet’s theorem, 347 discrete logarithm, 88 discriminant, 108 division algorithm, divisor, divisor function, 231, 405, 431 double coset, 73 double dual, 129 dual group, 127 eigenvalue, 146 eigenvector, 146 Eisenstein series, 453 equivalent polynomials, 73 Euclid’s lemma, 26 Euclid’s theorem, 33 Euclidean algorithm, 18 length, 18 Euler phi function, 54, 57, 227 Euler product, 330 Euler’s constant, 213 Euler’s theorem, 67 evaluation map, 85 even character, 326 even function, 401 eventually coincide, 397 exactly divide, 27 exponent, 83 exponential congruence, 97 factorization, 234 Fermat prime, 36, 107 Fermat’s last theorem, 183, 185 Fermat’s little theorem, 68 Fermat’s theorem, 407 Fibonacci numbers, 23 field, 49 floor function, xi formal power series, 205 Fourier transform, 135, 160 fractional part, 29, 206 Index Frobenius problem, 39 fundamental theorem of arithmetic, 26 Gauss sum, 152 classical, 153 Gauss’s lemma, 103 Gaussian integer, 453 Gaussian set, 103 generalized von Mangoldt function, 290 generating function, 483 generator, 70 greatest common divisor, 12 polynomial, 91 group, 10 group character, 126 group of units, 49 Haar measure, 134 Heisenberg group, 16 Hensel’s lemma, 116 homomorphism group, 13 ring, 48 Hypothesis H, 288 ideal, 90, 171 image, 16 incongruent, 45 integer part, xi, 28, 206 integer-valued polynomial, 356, 357 integral domain, 174 integral operator, 146 invertible class, 55 involution, 403 isomorphism, 13 Jacobi symbol, 114 Jacobi’s theorem, 431 k-abundant number, 260 kernel, 16 Kneser’s theorem, 397 511 L-function, 330 -function, 275 Lagrange’s theorem, 69, 355 Lam´e’s theorem, 25 lattice point, 233 Laurent polynomial, 181 leading coefficient, 84 least common multiple, 28 least nonnegative residue, 46 Legendre symbol, 101, 153 Leibniz formula, 119 lexicographic order, linear diophantine equation, 39 Liouville’s formulae, 402, 419, 420 Liouville’s function, 226 Ljunggren equation, 42 localization, 180 logarithmic derivative, 177 logarithmic integral, 298 lower asymptotic density, 256, 360, 482 m-adic representation, mathematical induction, xii, mean value, 206 Mersenne prime, 36, 107, 242 Mertens’s formula, 279 Mertens’s theorem, 276 middle binomial coefficient, 268 minimum principle, multiple, multiplicative character, 326 multiplicative function, 58, 217, 224, 430 multiplicatively closed, 179 Măobius function, 217 Măobius inversion, 218 nilpotent, 56, 172 norm L2 , 134 L∞ , 137 NSE, 367, 376 odd character, 326 512 Index odd function, 401 order, 68 group, 69 group element, 70 lexicographic, partial, 10 total, 10 order modulo m, 83 order of magnitude, xii, 273 orthogonality relations, 129, 130, 327 p-adic value, 27 p-group, 121 pairing, 129 pairwise relatively prime, 13 partial fractions, 462 partial order, 10 partial quotients, 19 partial summation, 211 partition, 455 partition function, 455 perfect number, 241 plaintext, 76 pointwise product, 201 pointwise sum, 201 polynomial, 84 congruent, 90 degree, 84 derivative, 116 monic, 84 root, 85 zero, 85 power, 189 power residue, 98 powerful number, 32, 187 prime ideal, 171 prime number, 25 prime number race, 351 prime number theorem, 274, 289 primitive abundant number, 260 primitive root, 84 primitive set, 255 principal character, 151, 326 principal ideal, 171 principal ring, 171 product ideal, 175 projective space, 15 pseudoprime, 75 public key cryptosystem, 76, 78 quadratic form, 108, 404 quadratic nonresidue, 98, 101 quadratic reciprocity law, 109 quadratic residue, 98, 100 quotient, quotient field, 176, 180 quotient group, 73 radical, 30, 172, 218 of a polynomial, 173 of an integer, 172 radical ideal, 172 Ramanujan-Nagell equation, 42 real character, 326 reduced set of residues, 54 reflexive relation, relatively prime, 13 remainder, representation function, 367 residue class, 46 Riemann hypothesis, 323, 351 Riemann zeta function, 221, 335 ring, 48 ring of formal power series, 205 ring of fractions, 180 root of unity, 11 RSA cryptosystem, 79 secret key cryptosystem, 77 Selberg’s formula, 293, 294 set of multiples, 255 Shnirel’man density, 359 Shnirel’man’s addition theorem, 363 sieve of Eratosthenes, 34 simple continued fraction, 19 spectrum, 171 square-free integer, 32, 217 stable basis, 359 Index standard factorization, 27 subgroup, 11 sum function, 206 sumset, 121, 361 support, 137, 291 tauberian theorem, 486 Taylor’s formula, 119 ternary quadratic form, 405 theta function, 453 total order, 10 totient function, 54 trace of a matrix, 144 transitive relation, 10 translation invariant, 134 translation operator, 139, 146 twin primes, 31, 287 unimodal, 206, 268, 474 unit, 48 upper asymptotic density, 256, 482 von Mangoldt function, 223, 276 generalized, 290 Waring’s problem, 355 for polynomials, 356 weight function, 375 weighted set, 375 Wieferich prime, 187 Wieferich’s theorem, 355 Wilson’s theorem, 53 zero set, 173 513 Graduate Texts in Mathematics (continuedfrom page ii) 62 KAROAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups 63 BoLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol 12nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed 66 WATERHOUSE Introduction to Affine Group Schemes 67 SERRE Local Fields 68 WEIDMANN Linear Operators in Hilbert Spaces 69 LANG Cyclotomic Fields XL 70 MASSEY Singular Homology Theory 71 FARKAS/KRA Riemann Surfaces 2nd ed 72 STILL WELL Classical Topology and 93 94 DUBROVIN/FOMENKO/NOVIKOV Modem Geometry—Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups 95 SHIRYAEV Probabihty 2nd ed 96 CONWAY A Course in Functional Analysis 2nd ed KOBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed 97 98 BR6CKER/TOM DIECK Representations of 99 Compact Lie Groups GROVE/BENSON Finite Reflection Groups 2nd ed 100 BERG/CHRISTENSEN/RESSEL Harmonic Combinatorial Group Theory 2nd ed 73 HuNGERFORD Algebra 74 DAVENPORT Multiplicative Number Theory 2nd ed 75 HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras 76 IiTAKA Algebraic Geometry 77 HECKE Lectures on the Theory of Algebraic Numbers Analysis on Semigroups: Theory of Positive Definite and Related Functions 101 EDWARDS Galois Theory 102 VARADARAJAN Lie Groups, Lie Algebras and Their Representations 103 LANG Complex Analysis 3rd ed 78 105 106 79 80 81 82 83 84 85 86 87 88 89 90 91 92 BURRIS/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FORSTER Lectures on Riemann Surfaces BOTT/TU Differential Forms in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELAND/ROSEN A Classical Introduction to Modem Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERCE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BR0NDSTED An Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups DiESTEL Sequences and Series in Banach Spaces 104 DUBROVIN/FOMENKO/NOVIKOV Modem 107 108 109 110 111 112 113 114 115 Geometry—Methods and Applications Part II hmG.SL2(R) SILVERMAN The Arithmetic of Elliptic Curves OLVER Applications of Lie Groups to Differential Equations 2nd ed RANGE Holomorphic Functions and Integral Representations in Several Complex Variables LEHTO Univalent Functions and TeichmuUer Spaces LANG Algebraic Niunber Theory HUSEMOLLER Elliptic Curves LANG Elliptic Functions KARATZAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed KOBLITZ A Course in Number Theory and Cryptography 2nd ed BERGER/GOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces 116 KELLEY/SRINIVASAN Measure and Integral Vol I 117 SERRE Algebraic Groups and Class Fields 118 PEDERSEN Analysis Now 119 ROTMAN An Introduction to Algebraic Topology 120 ZlEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAUS/HERMES et al Numbers Readings in Mathematics 124 DUBROVIN/FOMENKO/NOVIKOV Modem 125 126 127 128 129 130 131 132 133 134 135 Geometry—Methods and Applications Fart III BERENSTEIN/GAY Complex Variables: An Introduction BOREL Linear Algebraic Groups 2nd ed MASSEY A Basic Course in Algebraic Topology RAUCH Partial Differential Equations FULTON/HARRIS Representation Theory: A First Course Readings in Mathematics DODSON/POSTON Tensor Geometry LAM A First Coiu'se in Noncommutative Rings BEARDON Iteration of Rational Functions HARRIS Algebraic Geometry: A First Course ROMAN Coding and Information Theory ROMAN Advanced Linear Algebra 136 ADKINSAVEINTRAUB Algebra: An Approach via Module Theory 137 AXLER/BOURDON/RAMEY Harmonic Function Theory 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKER/WEISPFENNING/KREDEL GrSbner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNIS/FARB Noncommutative Algebra 145 ViCK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic A^-Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 ElSENBUD Commutative Algebra with a View Toward Algebraic Geometry 151 SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWN/PEARCY An Introduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAVIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEIN/ERDELYI Polynomials and Polynomial Inequalities 162 ALPERIN/BELL Groups and Representations 163 DIXON/MORTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An Introduction to Knot Theory 176 LEE Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 CLARKE/LEDYAEV/STERNAVOLENSKI Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KRESS Numerical Analysis 182 WALTER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOLLOBAS Modem Graph Theory 185 COX/LITTLE/0'SHEA Using Algebraic Geometry 186 RAMAKRISHNANA'ALENZA Fourier Analysis on Number Fields 187 HARRIS/MORRISON Moduli of Curves 188 GOLDBLATT Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 ESMONDE/MURTY Problems in Algebraic Number Theory 191 LANG Fundamentals of Differential Geometry 192 HIRSCH/LACOMBE Elements of Functional Analysis 193 COHEN Advanced Topics in Computational Number Theory 194 ENGEL/NAGEL One-Parameter Semigroups for Linear Evolution Equations 195 NATHANSON Elementary Methods in Number Theory ... Homotopy Theory (continued after index) Melvyn B Nathanson Elementary Methods in Number Theory Springer Melvyn B Nathanson Department of Mathematics Lehman College (CUNY) Bronx, NY 10468 USA nathansn@alpha.lehman.cuny.edu... in your head faster and faster until you blow up with the answer KABOOM!!! Then you sit back down and begin the next problem Alexander Nathanson [99] This book, Elementary Methods in Number Theory,. .. A and B be nonempty sets of integers and d ∈ Z We define the sumset A + B = {a + b : a ∈ A, b ∈ B} , the difference set A − B = {a − b : a ∈ A, b ∈ B} , the product set AB = {ab : a ∈ A, b ∈ B} , and