Introduction to Number Theory

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L. -K. Hua
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'1. The Factorization of Integers.- 1.1 Divisibility.- 1.2 Prime Numbers and Composite Numbers.- 1.3 Prime Numbers.- 1.4 Integral Modulus.- 1.5 The Fundamental Theorem of Arithmetic.- 1.6 The Greatest Common Factor and the Least Common Multiple.- 1.7 The Inclusion-Exclusion Principle.- 1.8 Linear Indeterminate Equations.- 1.9 Perfect Numbers.- 1.10 Mersenne Numbers and Fermat Numbers.- 1.11 The Prime Power in a Factorial.- 1.12 Integral Valued Polynomials.- 1.13 The Factorization of Polynomials.- Notes.- 2. Congruences.- 2.1 Definition.- 2.2 Fundamental Properties of Congruences.- 2.3 Reduced Residue System.- 2.4 The Divisibility of 2p-1-1 by p2.- 2.5 The Function ?(m).- 2.6 Congruences.- 2.7 The Chinese Remainder Theorem.- 2.8 Higher Degree Congruences.- 2.9 Higher Degree Congruences to a Prime Power Modulus.- 2.10 Wolstenholme's Theorem.- 3. Quadratic Residues.- 3.1 Definitions and Euler's Criterion.- 3.2 The Evaluation of Legendre's Symbol.- 3.3 The Law of Quadratic Reciprocity.- 3.4 Practical Methods for the Solutions.- 3.5 The Number of Roots of a Quadratic Congruence.- 3.6 Jacobi's Symbol.- 3.7 Two Terms Congruences.- 3.8 Primitive Roots and Indices.- 3.9 The Structure of a Reduced Residue System.- 4. Properties of Polynomials.- 4.1 The Division of Polynomials.- 4.2 The Unique Factorization Theorem.- 4.3 Congruences.- 4.4 Integer Coefficients Polynomials.- 4.5 Polynomial Congruences with a Prime Modulus.- 4.6 On Several Theorems Concerning Factorizations.- 4.7 Double Moduli Congruences.- 4.8 Generalization of Fermat's Theorem.- 4.9 Irreducible Polynomials mod p.- 4.10 Primitive Roots.- 4.11 Summary.- 5. The Distribution of Prime Numbers.- 5.1 Order of Infinity.- 5.2 The Logarithm Function.- 5.3 Introduction.- 5.4 The Number of Primes is Infinite.- 5.5 Almost All Integers are Composite.- 5.6 Chebyshev's Theorem.- 5.7 Bertrand's Postulate.- 5.8 Estimation of a Sum by an Integral.- 5.9 Consequences of Chebyshev's Theorem.- 5.10 The Number of Prime Factors of n.- 5.11 A Prime Representing Function.- 5.12 On Primes in an Arithmetic Progression.- Notes.- 6. Arithmetic Functions.- 6.1 Examples of Arithmetic Functions.- 6.2 Properties of Multiplicative Functions.- 6.3 The Möbius Inversion Formula.- 6.4 The Möbius Transformation.- 6.5 The Divisor Function.- 6.6 Two Theorems Related to Asymptotic Densities.- 6.7 The Representation of Integers as a Sum of Two Squares.- 6.8 The Methods of Partial Summation and Integration.- 6.9 The Circle Problem.- 6.10 Farey Sequence and Its Applications.- 6.11 Vinogradov's Method of Estimating Sums of Fractional Parts.- 6.12 Application of Vinogradov's Theorem to Lattice Point Problems.- 6.13 ?-results.- 6.14 Dirichlet Series.- 6.15 Lambert Series.- Notes.- 7. Trigonometric Sums and Characters.- 7.1 Representation of Residue Classes.- 7.2 Character Functions.- 7.3 Types of Characters.- 7.4 Character Sums.- 7.5 Gauss Sums.- 7.6 Character Sums and Trigonometric Sums.- 7.7 From Complete Sums to Incomplete Sums.- 7.8 Applications of the Character Sum $$sumlimits_{x = 1}^p {left( {frac{{x^2 + ax + b}}{p}}
ight)} $$.- 7.9 The Problem of the Distribution of Primitive Roots.- 7.10 Trigonometric Sums Involving Polynomials.- Notes.- 8. On Several Arithmetic Problems Associated with the Elliptic Modular Function.- 8.1 Introduction.- 8.2 The Partition of Integers.- 8.3 Jacobi's Identity.- 8.4 Methods of Representing Partitions.- 8.5 Graphical Method for Partitions.- 8.6 Estimates for p(n).- 8.7 The Problem of Sums of Squares.- 8.8 Density.- 8.9 A Summary of the Problem of Sums of Squares.- 9. The Prime Number Theorem.- 9.1 Introduction.- 9.2 The Riemann ?-Function.- 9.3 Several Lemmas.- 9.4 A Tauberian Theorem.- 9.5 The Prime Number Theorem.- 9.6 Selberg's Asymptotic Formula.- 9.7 Elementary Proof of the Prime Number Theorem.- 9.8 Dirichlet's Theorem.- Notes.- 10. Continued Fractions and Approximation Meth
to Number Theory Translated from the Chinese by Peter Shiu With 14 Figures Springer-Verlag Berlin Heidelberg New York 1982 HuaLooKeng Institute of Mathematics Academia Sinica Beijing The People's Republic of China PeterShlu Department of Mathematics University of Technology Loughborough Leicestershire LE 11 3 TU United Kingdom ISBN -13 : 978-3-642-68132-5 e-ISBN -13 : 978-3-642-68130-1 DOl: 10.1007/978-3-642-68130-1 Library of Congress Cataloging in Publication Data. Hua, Loo-Keng, 1910 -. Introduc tion to number theory. Translation of: Shu lun tao yin. Bibliography: p. Includes index. 1. Numbers, Theory of. I. Title. QA241.H7513.5 12'.7.82-645. ISBN-13:978-3-642-68132-5 (U.S.). AACR2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustra tions, broadcasting, reproductiOli by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "VerwertungsgeselIschaft Wort", Munich. © Springer-Verlag Berlin Heidelberg 1982 Softcover reprint of the hardcover 1st edition 1982 Typesetting: Buchdruckerei Dipl.-Ing. Schwarz' Erben KG, Zwettl. 214113140-5432 I 0 Preface to the English Edition The reasons for writing this book have already been given in the preface to the original edition and it suffices to append a few more points.

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