基本數論

基本數論
定價：276 元
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內容簡介

Our intention in writing this book is to give an elementary introduction to number theory which does not demand a great deal of mathematical background or maturity from the reader, and which can be read and understood with no extra assistance. Our first three chapters are based almost entirely on A-level mathematics, while the next five require little else beyond some elementary group theory. It is only in the last three chapters, where we treat more advanced topics, including recent developments, that we require greater mathematical background; here we use some basic ideas which students would expect to meet in the first year or so of a typical undergraduate course in mathematics. Throughout the book, we have attempted to explain our arrangements as fully and as clearly as possible, with plenty of worked examples and with outline solutions for all exercises.
 

目錄

Notes to the Reader
1. Divisibility
　1.1　Divisors
　1.2　Bezout』s identity
　1.3　Least common multiples
　1.4　Linear Diophantine equations
　1.5　Supplementary exercises
2. Prime Numbers
　2.1 Prime numbers and prime-power factorisations
　2.2 Distribution of primes
　2.3 Fermat and Mersenne primes
　2.4 Primality-testing and factorisation
　2.5 Supplementary exercises
3. Congruences
　3.1 Modular arithmetic
　3.2 Linear congruences
　3.3 Simultaneous linear congruences
　3.4 Simultaneous non-linear congruences
　3.5 An extension of the Chinese Remainder Theorem
　3.6 Supplementary exercises
4. Congruences with a Prime-power Modulus
　4.1 The arithmetic of Zp
　4.2 Pseudoprimes and Carmiehael numbers
　4.3 Solving congruences mod (pe)
　4.4 Supplementary exercises
5. EulerTs Function
　5.1 Units
　5.2 Euler’’s function
　5.3 Applications of Euler’’s function
　5.4 Supplementary exercises
6. The Group of Units
　6.1 The group Un
　6.2 Primitive roots
　6.3 The group Ups, where p is an odd prime
　6.4 The group U2
　6.5 The existence of primitive roots
　6.6 Applications of primitive roots
　6.7 The algebraic structure of Un
　6.8 The universal exponent
　6.9 Supplementary exercises
7. Quadratic Residues
　7.1 Quadratic congruences
　7.2 The group of quadratic residues
　7.3 The Legendre symbol
　7.4 Quadratic reciprocity
　7.5 Quadratic residues for prime-power moduli
　7.6 Quadratic residues for arbitrary moduli
　7.7 Supplementary exercises
8. Arithmetic Functions
　8.1 Definition and examples
　8.2 Perfect numbers
　8.3 The MSbius Inversion Formula
　8.4 An application of the M6bius Inversion Formula
　8.5 Properties of the M6bius function
　8.6 The Dirichlet product
　8.7 Supplementary exercises
9. The Riemann Zeta Function
　9.1 Historical background
　9.2 Convergence
　9.3 Applications to prime numbers
　　……
10. Sums of Squares
11. Fermat』s Last Theorem
Appendix A. Induction and Well-ordering
Appendix B. Groups, Rings and Fields
Appendix C. Convergence
Appendix D. Table of Primes pSolutions to Exercises
Bibliography
Index of symbols
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