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This page intentionally left blank Now into its eighth edition and with additional material onprimality testing, written by J. H. Davenport, The HigherArithmetic introduces concepts and theorems in a way that doesnot require the reader to have an in-depth knowledge of thetheory of numbers but also touches upon matters of deepmathematical significance. A companion website(www.cambridge.org/davenport) provides more details of thelatest advances and sample code for important algorithms.Reviews of earlier editions:‘ .the well-known and charming introduction to number theory .can be recommended both for independent study and as areference text for a general mathematical audience.’European Maths Society Journal‘Although this book is not written as a textbook but rather as awork for the general reader, it could certainly be used as atextbook for an undergraduate course in number theory and, inthe reviewer’s opinion, is far superior for this purpose to anyother book in English.’Bulletin of the American Mathematical Society THE HIGHERARITHMETICAN INTRODUCTION TOTHE THEORY OF NUMBERSEighth editionH. DavenportM.A., SC.D., F.R.S.late Rouse Ball Professor of Mathematicsin the University of Cambridge andFellow of Trinity CollegeEditing and additional material byJames H. Davenport CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São PauloCambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UKFirst published in print formatISBN-13 978-0-521-72236-0ISBN-13 978-0-511-45555-1© The estate of H. Davenport 20082008Information on this title: www.cambridge.org/9780521722360This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any partmay take place without the written permission of Cambridge University Press.Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.Published in the United States of America by Cambridge University Press, New Yorkwww.cambridge.orgeBook (EBL)paperback CONTENTSIntroductionpage viiiI Factorization and the Primes11. The laws of arithmetic 12. Proof by induction 63. Prime numbers 84. The fundamental theorem of arithmetic 95. Consequences of the fundamental theorem 126. Euclid’s algorithm 167. Another proof of the fundamental theorem 188. A property of the H.C.F 199. Factorizing a number 2210. The series of primes 25II Congruences311. The congruence notation 312. Linear congruences 333. Fermat’s theorem 354. Euler’s function φ(m) 375. Wilson’s theorem 406. Algebraic congruences 417. Congruences to a prime modulus 428. Congruences in several unknowns 459. Congruences covering all numbers 46v viContentsIII Quadratic Residues491. Primitive roots 492. Indices 533. Quadratic residues 554. Gauss’s lemma 585. The law of reciprocity 596. The distribution of the quadratic residues 63IV Continued Fractions681. Introduction 682. The general continued fraction 703. Euler’s rule 724. The convergents to a continued fraction 745. The equation ax − by = 1 776. Infinite continued fractions 787. Diophantine approximation 828. Quadratic irrationals 839. Purely periodic continued fractions 8610. Lagrange’s theorem 9211. Pell’s equation 9412. A geometrical interpretation of continuedfractions 99V Sums of Squares1031. Numbers representable by two squares 1032. Primes of the form 4k + 1 1043. Constructions for x and y 1084. Representation by four squares 1115. Representation by three squares 114VI Quadratic Forms1161. Introduction 1162. Equivalent forms 1173. The discriminant 1204. The representation of a number by a form 1225. Three examples 1246. The reduction of positive definite forms 1267. The reduced forms 1288. The number of representations 1319. The class-number 133 ContentsviiVII Some Diophantine Equations1371. Introduction 1372. The equation x2+ y2= z21383. The equation ax2+ by2= z21404. Elliptic equations and curves 1455. Elliptic equations modulo primes 1516. Fermat’s Last Theorem 1547. The equation x3+ y3= z3+ w31578. Further developments 159VIII Computers and Number Theory1651. Introduction 1652. Testing for primality 1683. ‘Random’ number generators 1734. Pollard’s factoring methods 1795. Factoring and primality via elliptic curves 1856. Factoring large numbers 1887. The Diffie–Hellman cryptographic method 1948. The RSA cryptographic method 1999. Primality testing revisited 200Exercises 209Hints 222Answers 225Bibliography 235Index 237 INTRODUCTIONThe higher arithmetic, or the theory of numbers, is concerned with theproperties of the natural numbers 1, 2, 3, These numbers must haveexercised human curiosity from a very early period; and in all the recordsof ancient civilizations there is evidence of some preoccupation with arith-metic over and above the needs of everyday life. But as a systematic andindependent science, the higher arithmetic is entirely a creation of moderntimes, and can be said to date from the discoveries of Fermat (1601–1665).A peculiarity of the higher arithmetic is the great difficulty which hasoften been experienced in proving simple general theorems which hadbeen suggested quite naturally by numerical evidence. ‘It is just this,’ saidGauss, ‘which gives the higher arithmetic that magical charm which hasmade it the favourite science of the greatest mathematicians, not to men-tion its inexhaustible wealth, wherein it so greatly surpasses other parts ofmathematics.’The theory of numbers is generally considered to be the ‘purest’ branchof pure mathematics. It certainly has very few direct applications toother sciences, but it has one feature in common with them, namely theinspiration which it derives from experiment, which takes the form of test-ing possible general theorems by numerical examples. Such experiment,though necessary in some form to progress in every part of mathematics,has played a greater part in the development of the theory of numbers thanelsewhere; for in other branches of mathematics the evidence found in thisway is too often fragmentary and misleading.As regards the present book, the author is well aware that it will not beread without effort by those who are not, in some sense at least, mathe-maticians. But the difficulty is partly that of the subject itself. It cannot beevaded by using imperfect analogies, or by presenting the proofs in a wayviii 123doc.vn