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    Number Theory
    Number Theory
    Number Theory
    Ebook395 pages3 hours

    Number Theory

    By George E. Andrews

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    • L-Functions

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    About this ebook

    Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.
    In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.
    Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory.
    Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples. Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated.

    LanguageEnglish
    PublisherDover Publications
    Release dateApr 30, 2012
    ISBN9780486135106
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      Book preview

      Number Theory - George E. Andrews

      NUMBER THEORY

      GEORGE E. ANDREWS

      Evan Pugh Professor of Mathematics

      Pennsylvania State University

      DOVER PUBLICATIONS, INC.

      NEW YORK

      Copyright

      Copyright © 1971 by W.B. Saunders Company.

      Bibliographical Note

      This Dover edition, first published in 1994, is an unabridged and corrected republication of the work first published by W.B. Saunders Company, Philadelphia, in 1971.

      Library of Congress Cataloging-in-Publication Data

      Andrews, George E., 1938–

      Number theory / George E. Andrews. — 1st Dover ed.

      p. cm.

      Originally published: Philadelphia : Saunders, 1971.

      ISBN-13: 978-0-486-68252-5 (pbk.)

      ISBN-10: 0-486-68252-8 (pbk.)

      1. Number theory. I. Title.

      Manufactured in the United States by Courier Corporation

      68252811

      www.doverpublications.com

      To Joy and Amy

      NOTE ON THE COVER

      , the golden ratio. The ancient Egyptians may have used this ratio in the construction of pyramids. The ratio recurs often in number theory; for example,

      where D2(n) and D2′(n) are the partition functions occurring in the Rogers-Ramanujan identities, and Fn is the nth Fibonacci number.

      PREFACE

      Most mathematics majors first encounter number theory in courses on abstract algebra, for which number theory provides numerous examples of algebraic systems, such as finite groups, rings, and fields. The instructor of undergraduate number theory thus faces a predicament. He must interest advanced mathematics students, who have previously studied congruence’s and the fundamental theorem of arithmetic, as well as other students, mostly from education and liberal arts, who usually need a careful exposition of these basic topics.

      To interest a class of students whose backgrounds are so divergent, this text offers a combinatorial approach to elementary number theory. The rationale for this point of view is perhaps best summarized by Herbert Riser in Combinatorial Mathematics: … combinatory and number theory are sister disciplines. They share a certain inter section of common knowledge and each genuinely enriches the other.* In studying number theory from a combinatorial perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems (the proofs of Theorems 1–3, 3–4, 3–5, 6–1, 6–2, 6–3, 7–6, 8–4, and 9–4 rely mainly on simple combinatorial reasoning). Number theory and combinatory are combined in Chapters 10 through 15 to aid in the discovery and proof of theorems.

      Two aspects of the text require preliminary discussion. First, Section 5 of Chapter 3 is critical to the whole work. This section illustrates both the value of numerical examples in number theory and the role of computers in obtaining such examples. The accompanying exercises provide opportunities for constructing numerical tables, with or without a computer. Subsequent chapters may then be introduced to advantage by allowing students to report on conjectures they derive from relevant numerical tables. When students are thus actively involved, theorems will seem natural and well motivated.

      Second, in Chapters 12, 13, and 14, the student will encounter partitions, a topic in additive number theory. Too often, one obtains from number theory texts the impression that each topic has been thoroughly developed. The problems offered in such texts are either solved or unsolvable; at best, the student is invited to work a few peripheral problems. In this book, Chapter 12 attempts to communicate the excitement of the mathematical chase by devising a procedure for forming conjectures in partition theory. The exercises at the end of Chapter 12 provide the student with a number of opportunities for discovering theorems himself. Chapters 13 and 14 develop techniques in the application of generating functions to partition theory so that the student can prove some of the conjectures he made in Chapter 12. In presenting Chapter 14, the instructor should assign the exercises at the end prior to beginning lectures, in order to avoid the unmotivated presentation of complicated manipulations of series and products; through this procedure, the student is led to appreciate the relation between the exercises and the steps in the proof of the Rogers-Ramanujan identities and of Schur’s Theorem.

      Many people have aided me in preparing this book. I express particular thanks to the students in my class of Spring Term, 1970, at Penn State, who were taught from the completed text and who generously offered valuable suggestions. Professors H. L. Alder, G. L. Alexanderson, and G. Piranian read the entire manuscript and made many useful contributions. Carlos Puig and George Fleming of W. B. Saunders have skillfully guided the process of publication.

      Finally I pay tribute to my wife, Joy, who has been the most important helper in the creation of this book; at each stage her encouragement, intelligence, and energy have added significant value. She has been immensely creative both in writing expository material and in facilitating the communication of ideas to students. Without her aid, a mass of scribbled lecture notes would still be just that.

      For certain classes where the instructor deems it wise to omit material requiring calculus, I recommend using all or part of the following: Chapters 1, 2, 3 (omit Sections 3-3 and 3-4), 4, 5, 6, 7, 8 (omit Section 8–2), 9, 10 (omit Section 10–2 except for a brief discussion of Corollary 10–1), 11 (omit Section 11–2 save for a summary of the results), 12, and 15 (up to Definition 15-1).

      George E. Andrews

      *Ryser, Herbert J., Combinatorial Mathematics (Cams Monograph No. 14). Mathematical Association of America, 1963.

      CONTENTS

      Part IMULTIPUCATIVITY – DIVISIBILITY

      Chapter 1

      BASIS REPRESENTATION

      1–1Principle of Mathematical Induction

      1–2The Basis Representation Theorem

      Chapter 2

      THE FUNDAMENTAL THEOREM OF ARITHMETIC

      2–1Euclid’s Division Lemma

      2–2Divisibility

      2–3The Linear Diophantine Equation

      2–4The Fundamental Theorem of Arithmetic

      Chapter 3

      COMBINATORIAL AND COMPUTATIONAL NUMBER THEORY

      3–1Permutations and Combinations

      3–2Fermat’s Little Theorem

      3–3Wilson’s Theorem

      3–4Generating Functions

      3–5The Use of Computers in Number Theory

      Chapter 4

      FUNDAMENTALS OF CONGRUENCES

      4–1Basic Properties of Congruences

      4–2Residue Systems

      4–3Riffling

      Chapter 5

      SOLVING CONGRUENCES

      5–1Linear Congruences

      5–2The Theorems of Fermat and Wilson Revisited

      5–3The Chinese Remainder Theorem

      5–4Polynomial Congruences

      Chapter 6

      ARITHMETIC FUNCTIONS

      6–1Combinatorial Study of ϕ(n)

      6–2Formulae for d(n) and σ(n)

      6–3Multiplicative Arithmetic Functions

      6–4The Möbius Inversion Formula

      Chapter 7

      PRIMITIVE ROOTS

      7–1Properties of Reduced Residue Systems

      7–2Primitive Roots Modulo p

      Chapter 8

      PRIME NUMBERS

      8–1Elementary Properties of π(x)

      8–2Tchebychev’s Theorem

      8–3Some Unsolved Problems About Primes

      Part II  QUADRATIC CONGRUENCES

      Chapter 9

      QUADRATIC RESIDUES

      9–1Euler’s Criterion

      9–2The Legendre Symbol

      9–3The Quadratic Reciprocity Law

      9–4Applications of the Quadratic Reciprocity Law

      Chapter 10

      DISTRIBUTION OF QUADRATIC RESIDUES

      10–1Consecutive Residues and Nonresidues

      10–2Consecutive Triples of Quadratic Residues

      Part III  ADDITIVITY

      Chapter 11

      SUMS OF SQUARES

      11–1Sums of Two Squares

      11–2Sums of Four Squares

      Chapter 12

      ELEMENTARY PARTITION THEORY

      12–1Introduction

      12–2Graphical Representation

      12–3Euler’s Partition Theorem

      12–4Searching for Partition Identities

      Chapter 13

      PARTITION GENERATING FUNCTIONS

      13–1Infinite Products As Generating Functions

      13–2Identities Between Infinite Series and Products

      Chapter 14

      PARTITION IDENTITIES

      14–1History and Introduction

      14–2Euler’s Pentagonal Number Theorem

      14–3The Rogers-Ramanujan Identities

      14–4Series and Products Identities

      14–5Schur’s Theorem

      Part IV  GEOMETRIC NUMBER THEORY

      Chapter 15

      LATTICE POINTS

      15–1Gauss’s Circle Problem

      15–2Dirichlet’s Divisor Problem

      APPENDICES

      Appendix A

      Appendix B

      INFINITE SERIES AND PRODUCTS (Convergence and Rearrangement of Series and Products)

      (Maclaurin Series Expansion of Infinite Products)

      Appendix C

      DOUBLE SERIES

      Appendix D

      THE INTEGRAL TEST

      NOTES

      SUGGESTED READING

      BIBLIOGRAPHY

      HINTS AND ANSWERS TO SELECTED EXERCISES

      INDEX OF SYMBOLS

      INDEX

      PART I

      MULTIPLICATIVITY-DIVISIBILITY

      Part I is devoted to multiplicative problems; these are sometimes called divisibility problems, since division is the inverse of multiplication.

      The knowledge of divisibility that we gain in the first two chapters leads us to our first goal, the fundamental theorem of arithmetic, which discloses the important role of primes in multiplicative number theory. Chapter 3 introduces combinatorial techniques for solving important divisibility problems and answering other number-theoretic questions. In order that we can study divisibility problems in greater depth, Chapters 4 and 5 develop the theory of congruences. Chapter 6 discusses some of the important functions related to multiplication and division, for example the number d(n) of divisors of n and the sum σ(n) of the divisors of n. Our results on congruences are extended in Chapter 7. The final chapter of Part I is concerned with the distribution of primes.

      CHAPTER 1

      BASIS REPRESENTATION

      Our objective in this chapter is to prove the basis representation theorem (Theorem 1–3). First we need to understand the principle of mathematical induction, a tool indispensable in number theory.

      1–1PRINCIPLE OF MATHEMATICAL INDUCTION

      Let us try to answer the following question: What is the sum of all integers from one through n, for any positive integer n? If n = 1, the sum equals 1 because 1 is the only summand. The answer we seek is a formula that will enable us to determine this sum for each value of n without having to add the summands.

      Table 1–1 lists the sum Sn of the first n consecutive positive integers for values of n from 1 through 10. Notice that in each case Sn equals one-half the product of n and the next integer; that is,

      for n = 1, 2, 3, …, 10. Although this formula gives the correct value of Sn for the first ten values of n, we cannot be sure that it holds for n greater than 10.

      To construct Table 1–1, we do not need to compute Sn each time by adding the first n positive integers. Having obtained values of Sn for n less than or equal to some integer k, we can determine Sk + 1 simply by adding (k + 1) to Sk :

      TABLE 1-1: SUM Sn OF THE FIRST n CONSECUTIVE POSITIVE INTEGERS.

      This last approach suggests a way of verifying equation (1–1–1). Suppose we know that formula (1–1–1) is true for n ≤ k, where k is a positive integer. Then we know that

      and so

      that is,

      The last equation is the same as equation (1–1–1) except that n is replaced by k + 1.

      We have proved that if equation (1–1–1) holds for n ≤ k, then it holds for n = k + 1, and we have already verified that equation (1–1–1) holds for n = 1, 2, …, 10. Therefore, by the preceding argument, we conclude that equation (1–1–1) is also correct for n = 11. Since it holds for n = 1, 2, …, 11, the same process shows that it is correct for n = 12. Since it is true for n = 1, 2, …, 12, it is true for n = 13, and so on. We can describe the principle underlying the foregoing argument in various ways. The following formulation is the most appropriate for our purposes.

      PRINCIPLE OF MATHEMATICAL INDUCTION: A statement about integers is true for all integers greater than or equal to 1 if

      (i)it is true for the integer 1, and

      (ii)whenever it is true for all the integers 1, 2, …, k, then it is true for the integer k + 1.

      By a statement about integers we do not necessarily mean a formula. A sentence such as "n (n² – 1) (3n + 2) is divisible by 24" is also acceptable (see Exercise 17 of this section). The assumption that "the statement is true for n = 1, 2, …, k" will often be referred to as the induction hypothesis. Sometimes the role 1 plays in the Principle will be replaced by some other integer, say b; in such instances the principle of mathematical induction establishes the statement for all integers n ≥ b.

      Since we have shown that equation (1–1–1) fulfills conditions (i) and (ii), we conclude by the principle of mathematical induction that (1–1–1) is true for all integers n ≥ 1. We state this result as our first theorem.

      THEOREM 1–1:The sum of the first n positive integers is

      Our next theorem also illustrates the use of the principle of mathematical induction.

      THEOREM 1–2:If x is any real number other than 1, then

      is shorthand for A0 + A1 + A2 + … + An –1.

      PROOF: Again we proceed by mathematical induction. If n and (x – 1) / (x – 1) = 1. Thus the theorem is true for n = 1.

      , we find that

      COROLLARY 1–1:If m and n are positive integers and if m > 1, then n < mn.

      PROOF: Note that

      EXERCISES

      1.Prove that

      2.Prove that

      [Hint: Use Theorem 1–1.]

      3.Prove that

      4.Prove that

      5.Prove that

      6.Prove that

      7.Suppose that F1 = 1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, and in general Fn = Fn–1 + Fn–2 for n ≥ 3. (Fn is called the nth Fibonacci number.) Prove that

      In Exercises 8 through 16, Fn stands for the nth Fibonacci number. (See Exercise 7.)

      8.Prove that

      9.Prove that

      10.Prove that

      11.Prove that

      12.Prove that

      13.The Lucas numbers Ln are defined by the equations L1 = 1, and Ln = Fn + 1 + Fn –1 for each n ≥ 2. Prove that

      14.What is wrong with the following argument?

      "Assuming Ln = Fn for n = 1, 2, … , k, we see that

      Hence, by the principle of mathematical induction, Fn = Ln for each positive integer n."

      15.Prove that F2n = Fn Ln.

      16.Prove that

      17.Prove that n (n² – 1) (3n + 2) is divisible by 24 for each positive integer n.

      18.Prove that if n is an odd positive integer, then x + y is a factor of xn + yn. (For example, if n = 3, then x³ + y³ = (x + y)(x² – xy + y²).)

      1–2THE BASIS REPRESENTATION THEOREM

      Early in grade school, you learned to express the integers in terms of the decimal system of notation. The number ten is said to be the base for decimal notation, because the digits in any integer are coefficients of the progressive powers of 10.

      Example 1–1: In the decimal system, two hundred nine is written 209, which stands for

      Similarly, for four thousand one hundred twenty-nine we write 4129, which stands for

      We can likewise express integers in binary, or base two, notation. In this case the digits 0 and 1 are used as the coefficients of the progressive powers of 2.

      Example 1–2: In binary notation, we write twenty-three as 10111, which stands for

      and thirty-six has the form 100100, which stands for

      The basis representation theorem states that each integer greater than 1 can serve as a base for representing the positive

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