The Higher Arithmetic: An Introduction to the Theory of Numbers

by H. Davenport

• ISBN13: 9780521722360
• ISBN10: 0521722365
• Edition: 8th
• Format: Paperback
• Copyright: 2008-11-17
• Publisher: Cambridge University Press

Summary

The theory of numbers is generally considered to be the 'purest' branch of pure mathematics and demands exactness of thought and exposition from its devotees. It is also one of the most highly active and engaging areas of mathematics. Now into its eighth edition The Higher Arithmetic introduces the concepts and theorems of number theory in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical significance. Since earlier editions, additional material written by J. H. Davenport has been added, on topics such as Wiles' proof of Fermat's Last Theorem, computers and number theory, and primality testing. Written to be accessible to the general reader, with only high school mathematics as prerequisite, this classic book is also ideal for undergraduate courses on number theory, and covers all the necessary material clearly and succinctly.

Author Biography

H. Davenport, M.A., SC.D., F.R.S., late Rouse Ball Professor of Mathematics in the University of Cambridge and Fellow of Trinity College

Table of Contents

Introduction	p. viii
Factorization and the Primes	p. 1
The laws of arithmetic	p. 1
Proof by induction	p. 6
Prime numbers	p. 8
The fundamental theorem of arithmetic	p. 9
Consequences of the fundamental theorem	p. 12
Euclid's algorithm	p. 16
Another proof of the fundamental theorem	p. 18
A property of the H.C.F	p. 19
Factorizing a number	p. 22
The series of primes	p. 25
Congruences	p. 31
The congruence notation	p. 31
Linear congruences	p. 33
Fermat's theorem	p. 35
Euler's function [phi] (m)	p. 37
Wilson's theorem	p. 40
Algebraic congruences	p. 41
Congruences to a prime modulus	p. 42
Congruences in several unknowns	p. 45
Congruences covering all numbers	p. 46
Quadratic Residues	p. 49
Primitive roots	p. 49
Indices	p. 53
Quadratic residues	p. 55
Gauss's lemma	p. 58
The law of reciprocity	p. 59
The distribution of the quadratic residues	p. 63
Continued Fractions	p. 68
Introduction	p. 68
The general continued fraction	p. 70
Euler's rule	p. 72
The convergents to a continued fraction	p. 74
The equation ax - by = 1	p. 77
Infinite continued fractions	p. 78
Diophantine approximation	p. 82
Quadratic irrationals	p. 83
Purely periodic continued fractions	p. 86
Lagrange's theorem	p. 92
Pell's equation	p. 94
A geometrical interpretation of continued fractions	p. 99
Sums of Squares	p. 103
Numbers representable by two squares	p. 103
Primes of the form 4k + 1	p. 104
Constructions for x and y	p. 108
Representation by four squares	p. 111
Representation by three squares	p. 114
Quadratic Forms	p. 116
Introduction	p. 116
Equivalent forms	p. 117
The discriminant	p. 120
The representation of a number by a form	p. 122
Three examples	p. 124
The reduction of positive definite forms	p. 126
The reduced forms	p. 128
The number of representations	p. 131
The class-number	p. 133
Some Diphantine Equations	p. 137
Introduction	p. 137
The equation x[superscript 2] + y[superscript 2] = z[superscript 2]	p. 138
The equation ax[superscript 2] + by[superscript 2] = z[superscript 2]	p. 140
Elliptic equations and curves	p. 145
Elliptic equations modulo primes	p. 151
Fermat's Last Theorem	p. 154
The equation x[superscript 3] + y[superscript 3] = z[superscript 3] + w[superscript 3]	p. 157
Further developments	p. 159
Computers and Number Theory	p. 165
Introduction	p. 165
Testing for primality	p. 168
'Random' number generators	p. 173
Pollard's factoring methods	p. 179
Factoring and primality via elliptic curves	p. 185
Factoring large numbers	p. 188
The Diffie-Hellman cryptographic method	p. 194
The RSA cryptographic method	p. 199
Primality testing revisited	p. 200
Exercises	p. 209
Hints	p. 222
Answers	p. 225
Bibliography	p. 235
Index	p. 237
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