The higher arithmetic : an introduction to the theory of numbers /
H. Davenport.
- 6th edition
- Cambridge ; New York : Cambridge University Press, c1992.
- 217 p. : ill. ; 23 cm.
CONTENTS
Introduction I Factorization and the Primes 1. The laws of arithmetic 2. Proof by induction 3. Prime numbers 4. The fundamental theorem of arithmetic 5. Consequences of the fundamental theorem 6. Euclid’s algorithm 7. Another proof of the fundamental theorem 8. A property of the H.C.F 9. Factorizing a number 10. The series of primes
II Congruences 1. The congruence notation 2. Linear congruences 3. Fermat’s theorem 4. Euler’s function φ(m) 5. Wilson’s theorem 40 6. Algebraic congruences 7. Congruences to a prime modulus 8. Congruences in several unknowns 9. Congruences covering all numbers
III Quadratic Residues 1. Primitive roots 2. Indices 3. Quadratic residues 4. Gauss’s lemma 5. The law of reciprocity 6. The distribution of the quadratic residues
IV Continued Fractions 1. Introduction 2. The general continued fraction 3. Euler’s rule 4. The convergents to a continued fraction 5. The equation ax − by = 1 6. Infinite continued fractions 7. Diophantine approximation 8. Quadratic irrationals 9. Purely periodic continued fractions 10. Lagrange’s theorem 11. Pell’s equation 12. A geometrical interpretation of continued fractions
V Sums of Squares 1. Numbers representable by two squares 2. Primes of the form 4k + 1 3. Constructions for x and y 4. Representation by four squares 5. Representation by three squares
VI Quadratic Forms 1. Introduction 2. Equivalent forms 3. The discriminant 4. The representation of a number by a form 5. Three examples 6. The reduction of positive definite forms 7. The reduced forms 8. The number of representations 9. The class-number
VII Some Diophantine Equations 1. Introduction 2. The equation x2 + y2 = z2 3. The equation ax2 + by2 = z2 4. Elliptic equations and curves 5. Elliptic equations modulo primes 6. Fermat’s Last Theorem 7. The equation x3 + y3 = z3 + w3 8. Further developments
VIII Computers and Number Theory 1. Introduction 2. Testing for primality 3. ‘Random’ number generators 4. Pollard’s factoring methods 5. Factoring large numbers 6. The Diffie–Hellman cryptographic method 7. The RSA cryptographic method
Includes bibliographical references (p. 214-215) and index 216-217