基本数论

出版时间：2008-5  出版社：世界图书出版公司  作者：琼斯  页数：296  
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内容概要

　　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.

书籍目录

Notes to the Reader1. Divisibility1.1　Divisors1.2　Bezout’s identity1.3　Least common multiples1.4　Linear Diophantine equations1.5　Supplementary exercises2. Prime Numbers2.1 Prime numbers and prime-power factorisations2.2 Distribution of primes2.3 Fermat and Mersenne primes2.4 Primality-testing and factorisation2.5 Supplementary exercises3. Congruences3.1 Modular arithmetic3.2 Linear congruences3.3 Simultaneous linear congruences3.4 Simultaneous non-linear congruences3.5 An extension of the Chinese Remainder Theorem3.6 Supplementary exercises4. Congruences with a Prime-power Modulus4.1 The arithmetic of Zp4.2 Pseudoprimes and Carmiehael numbers4.3 Solving congruences mod （pe）4.4 Supplementary exercises5. EulerTs Function5.1 Units5.2 Euler's function5.3 Applications of Euler's function5.4 Supplementary exercises6. The Group of Units6.1 The group Un6.2 Primitive roots6.3 The group Ups， where p is an odd prime6.4 The group U26.5 The existence of primitive roots6.6 Applications of primitive roots6.7 The algebraic structure of Un6.8 The universal exponent6.9 Supplementary exercises7. Quadratic Residues7.1 Quadratic congruences7.2 The group of quadratic residues7.3 The Legendre symbol7.4 Quadratic reciprocity7.5 Quadratic residues for prime-power moduli7.6 Quadratic residues for arbitrary moduli7.7 Supplementary exercises8. Arithmetic Functions8.1 Definition and examples8.2 Perfect numbers8.3 The MSbius Inversion Formula8.4 An application of the M6bius Inversion Formula8.5 Properties of the M6bius function8.6 The Dirichlet product8.7 Supplementary exercises9. The Riemann Zeta Function9.1 Historical background9.2 Convergence9.3 Applications to prime numbers……10. Sums of Squares11. Fermat’s Last TheoremAppendix A. Induction and Well-orderingAppendix B. Groups， Rings and FieldsAppendix C. ConvergenceAppendix D. Table of Primes p