基本信息
- 原书名:An Introduction to the Theory of Numbers
- 原出版社: Oxford University Press
- 作者: (英)G. H. Hardy E. M. Wright
- 丛书名: 图灵原版数学·统计学系列
- 出版社:人民邮电出版社
- ISBN:9787115214270
- 上架时间:2009-10-28
- 出版日期:2009 年11月
- 开本:16开
- 页码:621
- 版次:6-1
- 所属分类:数学 > 代数,数论及组合理论 > 数论及应用
内容简介
作译者
E.M.Wright (1906-2005)(1906—2005)英国著名数学家,毕业于牛津大学,是G.H.Hardy的学生。生前担任英国名校阿伯丁大学校长多年。爱丁堡皇家学会会士、伦敦数学会会士。曾任Journal of Graph Theory和Zentralbtatt fur Mathematik的名誉主编。...
目录
1.1. Divisibility of integers 1
1.2. Prime numbers 2
1.3. Statement of the fundamental theorem of arithmetic 3
1.4. The sequence of primes 4
1.5. Some questions concerning primes 6
1.6. Some notations 7
1.7. The logarithmic function 9
1.8. Statement of the prime number theorem 10
II. THE SERIES OF PRIMES (2) 14
2.1. First proof of Euclid's second theorem 14
2.2. Further deductions from Euclid's argument 14
2.3. Primes in certain arithmetical progressions 15
2.4. Second proof of Euclid's theorem 17
2.5. Fermat's and Mersenne's numbers 18
2.6. Third proof of Euclid's theorem 20
2.7. Further results on formulae for primes 21
2.8. Unsolved problems concerning primes 23
2.9. Moduli of integers 23
2.10. Proof of the fundamental theorem or arithmetic 25
前言
This sixth edition contains a considerable expansion of the end-of-chapternotes. There have been many exciting developments since these were lastrevised, which are now described in the notes. It is hoped that these willprovide an avenue leading the interested reader towards current researchareas. The notes for some chapters were written with the generous help ofother authorities. Professor D. Masser updated the material on Chapters4 and 11, while Professor G.E. Andrews did the same for Chapter 19. Asubstantial amount of new material was added to the notes for Chapter 21by Professor T.D. Wooley, and a similar review of the notes for Chapter 24was undertaken by Professor R. Hans-Gill. We are naturally very gratefulto all of them for their assistance.
In addition, we have added a substantial new chapter, dealing with ellip-tic curves. This subject, which was not mentioned in earlier editions, hascome to be such a central topic in the theory of numbers that it was feltto deserve a full treatment. The material is naturally connected with theoriginal chapter on Dlophantlne Equations.
Finally, we have corrected a significant number of misprints in thefifth edition. A large number of correspondents reported typographical ormathematical errors, and we thank everyone who contributed in this way.
The proposal to produce this new edition originally came from ProfessorsJohn Maitland Wright and John Coates. We are very grateful for theirenthusiastic support.
D.R.H.-B...
J.H.S.
September, 2007
PREFACE TO THE FIFTH EDITION
The main changes in this edition are in the Notes at the end of each chapter.I have sought to provide up-to-date references for the reader who wishesto pursue a particular topic further and to present, both in the Notes and inthe text, a reasonably accurate account of the present state of knowledge.For this I have been dependent on the relevant sections of those invaluablepublications, the Zentralblatt and the Mathematical Reviews. But I wasalso greatly helped by several correspondents who suggested amendmentsor answered queries. I am especially grateful to Professors J. W. S. Casselsand H. Halberstam, each of whom supplied me at my request with a longand most valuable list of suggestions and references.
There is a new, more transparent proof of Theorem 445 and an account ofmy changed opinion about Theodorus'method in irrationals. To facilitatethe use of this edition for reference purposes, I have, so far as possible, keptthe page numbers unchanged. For this reason, I have added a short appendixon recent progress in some aspects of the theory of prime numbers, ratherthan insert the material in the appropriate places in the text.
E. M. W.
ABERDEEN
October 1978
PREFACE TO THE FIRST EDITION
time to change it to An introduction to arithmetic, a more novel and in someways a more appropriate title; but it was pointed out that this might lead tomisunderstandings about the content of the book.
A number of friends have helped us in the preparation of the book. Dr. H.Heilbronn has read all of it both in manuscript and in print, and his criticismsand suggestions have led to many very substantial improvements, the mostimportant of which are acknowledged in the text. Dr. H. S. A. Potter andDr. S. Wylie have read the proofs and helped us to remove many errors andobscurities. They have also checked most of the references to the literaturein the notes at the ends of the chapters. Dr. H. Davenport and Dr. R. Radohave also read parts of the book, and in particular the last chapter, which,after their suggestions and Dr. Heilbronn's, bears very little resemblanceto the original draft.
We have borrowed freely from the other books which are cataloguedon pp. 417-19 [pp. 596-9 in current 6th edn.], and especially from thoseof Landau and Perron. To Landau in particular we, in common with allserious students of the theory of numbers, owe a debt which we couldhardly overstate.
序言
I had the great good fortune to have a high school mathematics teacher whohad studied number theory. At his suggestion I acquired a copy of the fourth_edition of Hardy and Wnght's marvellous book An Introductton to the Theor), of Numbers. This, together with Davenport's The Higher Arithmetic,became my favourite introductory books in the suN ect Scourin~ th Da~eof the text for clues.about the Fermat problem (I was already obsessed) Ilearned for the first time about the real breadth of number theory. Only fourof the chapters m the middle of the book were about quadratic fields and Diophantine equations, and much of the rest of the material was new to me; Dlophantme geometry, round numbers, Dirichlet's theorem, continued fractions, quatemions, reciprocity... The list went on and on. ..
The book became a starting point for ventures into the different branches of !he subject. For me the first quest was to find out more about alge- braic number theory and Kummer's theory in particular. The more analytic parts did not have the same attractionthen and did not really catcn my lmagJn.atlon until I had learned some complex analysis. Only then could I appreciate the power.of the zeta function. However, the book was always there as a s!art~ng point which I could return to whenever I was intrigued by a new p~ece of theory, sometimes many years later. Part of the successf the book lay in its extensive notes and references which gave naviga- ~onal hints for the inexperienced mathematician. This part of the book has been updated and extended by Roger Heath-Brown so that a 21st- century-student can profit from more recent discoveries and texts. This isin the style of his wonderful commentary on Titchmarsh's The Theory ofthe Riemann Zeta Function. It will be an invaluable aid to the new readerbut it will also be a great pleasure to those who have read the book intheir youth, a bit like hearing the life stories of one's erstwhile schoolfriends.
A final chapter has been added giving an account of the theory of ellip-tic curves. Although this theory is not described in the original editions(except for a brief reference in the notes to ~ 13.6) it has proved to be crit-ical .in the study of Diophantine equations and of the Fermat equation inparticular. Through the Birch and Swinnerton-Dyer conjecture on the onehand and through the extraordinary link with the Fermat equation on theother It has become a central part of the number theorist's life. It evenplayed a central role in the effective resolution of a famous class numberproblem of Gauss. All this would have seemed absurdly improbable whenthe book was written. It is thus an appropriate ending for the new editionto have a lucid exposition of this theory by Joe Silverman. Of course it isonly a quick sketch of the theory and the reader will surely be tempted todevote many hours, if not the best part of a lifetime, to unravelling its manymysteries.
A.J.W. ...
January, 2008
媒体评论
——Mathematical Gazette(数学公报)..
“……一本非常重要的著作……它一定能继续保持长久、旺盛的生命力……”
——Mathematical Reviews(数学评论)...