# Book:H.E. Rose/A Course in Number Theory/Second Edition

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## H.E. Rose:

## Contents

## H.E. Rose: *A Course in Number Theory (2nd Edition)*

Published $\text {1994}$, **Oxford Science Publications**

- ISBN 0-19-852376-9.

### Subject Matter

### Contents

- Preface to Second Edition (
*Bristol, January $1994$*)

- Preface to First Edition (
*Bristol, February $1987$*)

- 1 Divisibility
- $1$ The Euclidean algorithm and unique factorization
- $2$ Prime numbers
- $3$ Problems $1$

- 2 Multiplicative Functions
- $1$ The Möbius and Euler functions
- $2$ Average order
- $3$ Problems $2$

- 3 Congruence Theory
- $1$ Definitions and linear congruences
- $2$ Nonlinear congruences and the theorems of Euler, Lagrange, and Chevalley
- $3$ Local versus global considerations
- $4$ Computation modulo $n$
- $5$ Problems $3$

- 4 Quadratic Residues
- $1$ The Legendre symbol
- $2$ Quadratic reciprocity
- $3$ Some further topics
- $4$ Problems $4$

- 5 Algebraic Topics
- $1$ Algebraic numbers and integers
- $2$ Primitive roots
- $3$ Characters
- $4$ Problems $5$

- 6 Sums of Squares and Gauss Sums
- $1$ Sums of squares
- $2$ Gauss and Jacobi sums
- $3$ The sign of the quadratic Gauss sum
- $4$ Problems $6$

- 7 Continued Fractions
- $1$ Basic properties
- $2$ Best approximation
- $3$ Pell's equation
- $4$ A set of real numbers modulo $1$
- $5$ Problems $7$

- 8 Transcendental Numbers
- $1$ Liouville's theorem and applications
- $2$ The Hermite and Lindemann theorems
- $3$ The Gelfond-Schneider theorem
- $4$ Problems $8$

- 9 Quadratic Forms
- $1$ Equivalence of forms
- $2$ Sums of three squares
- $3$ Representation by binary forms
- $4$ Algorithms for reduced forms
- $5$ Problems $9$

- 10 Genera and the Class Group
- $1$ The genus of a form
- $2$ Composition and the class group
- $3$ A formula for the class number
- $4$ Problems $10$

- 11 Partitions
- $1$ Elementary properties
- $2$ Jacobi's identity
- $3$ Estimates for $\map p n$
- $4$ Problems $11$

- 12 The Prime Numbers
- $1$ The results of Chebyshev and Bertrand
- $2$ Series involving primes
- $3$ Riemann zeta function
- $4$ Problems $12$

- 13 Two Major Theorems on the Primes
- $1$ Dirichlet's theorem
- $2$ PNT: preliminaries and Selberg's theorem
- $3$ PNT: the main proof
- $4$ Problems $13$

- 14 Diophantine Equations
- $1$ Legendre's theorem
- $2$ Fermat's last theorem
- $3$ Skolem's method
- $4$ Mordell's equation
- $5$ Problems $14$

- 15 Elliptic Curves: Basic Theory
- $1$ Geometric preliminaries
- $2$ Rational points on elliptic curves
- $3$ Mordell-Weil theorem
- $4$ Problems $15$

- 16 Elliptic Curves: Further Results and Applications
- $1$ Weierstrass equation
- $2$ Nagell-Lutz theorem
- $3$ Curves defined over finite fields
- $4$ Lenstra's factorization method
- $5$ $L$-functions for curves
- $6$ Problems $16$

- Answers and Hints to Problems
- Problems 1
- Problems 2
- Problems 3
- Problems 4
- Problems 5
- Problems 6
- Problems 7
- Problems 8
- Problems 9
- Problems 10
- Problems 11
- Problems 12
- Problems 13
- Problems 14
- Problems 15
- Problems 16

- Tables

- Bibliography

- Index of Notation

- General Index

## Further Editions

## Source work progress

- 1994: H.E. Rose:
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