基本信息
- 原书名：Friendly Introduction to Number Theory, (3rd Edition)
- 原出版社： Prentice Hall/Pearson
- 作者： （美）Joseph H.Silverman
- 丛书名： 经典原版书库
- 出版社：机械工业出版社
- ISBN：9787111196112
- 上架时间：2006-8-10
- 出版日期：2006 年8月
- 开本：16开
- 页码：434
- 版次：3-1
- 所属分类：数学 > 代数，数论及组合理论 > 数论及应用
教材
编辑推荐
我喜欢这本书。它讲解清晰，易于理解。用数值进行试验，用自己的方式从观察结果中猜 测，*后完成证明。
——Jurgen Bierbrauer，密歇根理工大学
本书每一章非常简短而且自成体系，很容易从中挑选我喜爱的主题。本书写作风格独特，书中提供了极佳的示例，以引出定理的叙述和证明。这种风格非常适合于数论的初级课程。
——Maureen Fenrick，明尼苏达州立大学曼凯托分校
内容简介
作译者
Joseph H．Silverman 拥有哈佛大学博士学位。他目前为布朗大学数学教授，之前曾任教于麻省理工学院和波士顿大学。1998年，他获得了美国数学会Steele奖的著述奖，获奖著作为《The Arithmetic of Elliptic Curves》和《Advanced Topics in the Arithmetic of Elliptic Curves》。...
目录
Preface.
Introduction
1. What Is Number Theory?
2. Pythagorean Triples
3. Pythagorean Triples and the Unit Circle
4. Sums of Higher Powers and Fermat’s Last Theorem
5. Divisibility and the Greatest Common Divisor
6. Linear Equations and the Greatest Common Divisor
7. Factorization and the Fundamental Theorem of Arithmetic
8. Congruences
9. Congruences, Powers, and Fermat’s Little Theorem
10. Congruences, Powers, and Euler’s Formula
11. Euler’s Phi Function and the Chinese Remainder Theorem
12. Prime Numbers
13. Counting Primes
14. Mersenne Primes
15. Mersenne Primes and Perfect Numbers8
16. Powers Modulo m and Successive Squaring
17. Computing kth Roots Modulo m
18. Powers, Roots, and “Unbreakable” Codes
Introduction
1. What Is Number Theory?
2. Pythagorean Triples
3. Pythagorean Triples and the Unit Circle
4. Sums of Higher Powers and Fermat’s Last Theorem
5. Divisibility and the Greatest Common Divisor
6. Linear Equations and the Greatest Common Divisor
7. Factorization and the Fundamental Theorem of Arithmetic
8. Congruences
9. Congruences, Powers, and Fermat’s Little Theorem
10. Congruences, Powers, and Euler’s Formula
11. Euler’s Phi Function and the Chinese Remainder Theorem
12. Prime Numbers
13. Counting Primes
14. Mersenne Primes
15. Mersenne Primes and Perfect Numbers8
16. Powers Modulo m and Successive Squaring
17. Computing kth Roots Modulo m
18. Powers, Roots, and “Unbreakable” Codes
前言
The 1990s saw a wave of calculus reform whose aim was to teach students to think for themselves and to solve substantial problems, rather than merely memorizing formulas and performing rote algebraic manipulations. This book has a similar, albeit somewhat more ambitious, goal; to lead you to think mathematically and to experience the thrill of independent intellectual discovery. Our chosen subject, Number Theory, is particularly well suited for this purpose. The natural numbers 1, 2, 3 .... satisfy a multitude of beautiful patterns and relationships, many of which can be discerned at a glance; others are so subtle that one marvels they were noticed at all. Experimentation requires nothing more than paper and pencil, but many false alleys beckon to those who make conjectures on too scanty evidence. It is only by rigorous demonstration that one is finally convinced that the numerical evidence reflects a universal truth. This book will lead you through the groves wherein lurk some of the brightest flowers of Number Theory, as it simultaneously encourages you to investigate, analyze, conjecture, and ultimately prove your own beautiful number theoretic results. .
This book was originally written to serve as a text for Math 42, a course created by Jeff Hoffstein at Brown University in the early 1990s. Math 42 was designed to attract nonscience majors, those with little interest in pursuing the standard calculus sequence, and to convince them to study some college mathematics. The intent was to create a course similar to one on,'say, "The Music of Mozart" or "Elizabethan Drama," wherein an audience is introduced to the overall themes and methodology of an entire discipline through the detailed study of a particular facet of the subject. Math 42 has been extremely successful, attracting both its intended audience and also scientifically oriented undergraduates interested in a change of pace from their large-lecture, cookbook-style courses.
The prerequisites for reading this book are few. Some facility with high schoolalgebra is required, and those who know how to program a computer will have fun generating reams of data and implementing assorted algorithms, but in truth the reader needs nothing more than a simple calculator. Concepts from calculus are mentioned in passing, but are not used in an essential way. However, and the reader is hereby forewarned, it is not possible to truly appreciate Number Theory without an eager and questioning mind and a spirit that is not afraid to experiment, to make mistakes and profit from them, to accept frustration and persevere to the ultimate triumph. Readers who are able to cultivate these qualities will find themselves richly rewarded, both in their study of Number Theory and their appreciation of all that life has to ofter.
Acknowledgments for the First Edition
There are many people I would like to thank for their assistance--Jeff Hoffstein, Karen Bender, and Rachel Pries for their pioneering work in Math 42; Bill Amend for kindly permitting me to use some of his wonderful FoxTrot cartoons; the creators of PARI for providing the ultimate in number theory computational power; Nick Fiori, Daniel Goldston, Rob Gross, Matt Holford, Alan Landman, Paul Lockhart, Matt Marcy, Patricia Pacelli, Rachel Pries (again), Michael Schlessinger, Thomas Shemanske, Jeffrey Stopple, Chris Towse, Roger Ware, Larry Washington, Yangbo Ye, and Karl Zimmerman for looking at the initial draft and offering invaluable suggestions; Michael Artin, Richard Guy, Marc Hindry, Mike Rosen, Karl Rubin, Ed Scheinerman, John Selfridge, and Sam Wagstaff for much helpful advice; and George Lobell and Gale Epps at Prentice Hall for their excellent advice and guidance during the publication process. ..
Finally, and most important, I want to thank my wife Susan and children Debby, Daniel, and Jonathan for their patience and understanding while this book was being written.
Acknowledgments for the Second Edition
I would like to thank all those who took the time to send me corrections and suggestions that were invaluable in preparing this second edition, including Arthur Baragar, Aaron Bertram, Nigel Boston, David Boyd, Seth Braver, Michael Catalano-Johnson, L. Chang, Robin Chapman, Miguel Cordero, John Cremona, Jim Delany, Lisa Fastenberg, Nicholas Fiori, Fumiyasu Funami, Jim Funderburk, Andrew Granville, Rob Gross, Shamita Dutta Gupta, Tom Hagedorn, Ron Jacobowitz, Jerry S. Kelly, Hershy Kisilevsky, Hendrik Lenstra, Gordon S. Lessells, Ken Levasseur, Stephen Lichtenbaum, Nidia Lopez Jerry Metzger, Jukka Pihko, Carl Pomerance, Rachel Pries, Ken Ribet, John Robeson, David Rohrlich, Daniel Silverman, Alfred Tang, and Wenchao Zhou.
Acknowledgments for the Third Edition
I would like to thank Jiro Suzuki for his beautiful translation of my book into Japanese. I would also like to thank all those who took the time to send me corrections and suggestions that were invaluable in preparing this third edition, including Bill Adams, Autumn Alden, Robert Altshuler, Avner Ash, Joe Auslander, Dave Benoit, Jfirgen Bierbrauer, Andrew Clifford, Keith Conrad, Sarah DeGooyer, Amartya Kumar Dutta, Laurie Fanning, Benji Fisher, Joe Fisher, Jon Graft, Eric Gutman, Edward Hinson, Bruce Hugo, Ole Jensen, Peter Kahn, Avinash Kalra, Jerry Kelly, Yukio Kikuchi, Amartya Kumar, Andrew Lenard, Sufatrio Liu, Troy Madsen, Russ Mann, Gordon Mason, Farley Mawyer, Mike McConnell, Jerry Metzger, Steve Paik, Nicole Perez, Dinakar Ramakrishnan, Cecil Rousseau, Marc Roth, Ehud Schreiber, Tamina Stephenson, Jiro Suzuki, James Tanton, James Tong, Chris Towse, Roger Turton, Fernando Villegas, and Chung Yi.
Email and Electronic Resources
All the people listed above have helped me to correct numerous mistakes and to greatly refine the exposition, but no book is ever free from error or incapable of being improved. I would be delighted to receive comments, good or bad, and corrections from my readers. You can send mail to me at jhs@math. brown. edu
Additional material, including an errata sheet, links to interesting number theoretic sites, and downloadable versions of various computer exercises, are available on the Friendly Introduction to Number Theory Home Page:
www. math. brown. edu/~ jhs / frint. html
Joseph H. Silverman ...
This book was originally written to serve as a text for Math 42, a course created by Jeff Hoffstein at Brown University in the early 1990s. Math 42 was designed to attract nonscience majors, those with little interest in pursuing the standard calculus sequence, and to convince them to study some college mathematics. The intent was to create a course similar to one on,'say, "The Music of Mozart" or "Elizabethan Drama," wherein an audience is introduced to the overall themes and methodology of an entire discipline through the detailed study of a particular facet of the subject. Math 42 has been extremely successful, attracting both its intended audience and also scientifically oriented undergraduates interested in a change of pace from their large-lecture, cookbook-style courses.
The prerequisites for reading this book are few. Some facility with high schoolalgebra is required, and those who know how to program a computer will have fun generating reams of data and implementing assorted algorithms, but in truth the reader needs nothing more than a simple calculator. Concepts from calculus are mentioned in passing, but are not used in an essential way. However, and the reader is hereby forewarned, it is not possible to truly appreciate Number Theory without an eager and questioning mind and a spirit that is not afraid to experiment, to make mistakes and profit from them, to accept frustration and persevere to the ultimate triumph. Readers who are able to cultivate these qualities will find themselves richly rewarded, both in their study of Number Theory and their appreciation of all that life has to ofter.
Acknowledgments for the First Edition
There are many people I would like to thank for their assistance--Jeff Hoffstein, Karen Bender, and Rachel Pries for their pioneering work in Math 42; Bill Amend for kindly permitting me to use some of his wonderful FoxTrot cartoons; the creators of PARI for providing the ultimate in number theory computational power; Nick Fiori, Daniel Goldston, Rob Gross, Matt Holford, Alan Landman, Paul Lockhart, Matt Marcy, Patricia Pacelli, Rachel Pries (again), Michael Schlessinger, Thomas Shemanske, Jeffrey Stopple, Chris Towse, Roger Ware, Larry Washington, Yangbo Ye, and Karl Zimmerman for looking at the initial draft and offering invaluable suggestions; Michael Artin, Richard Guy, Marc Hindry, Mike Rosen, Karl Rubin, Ed Scheinerman, John Selfridge, and Sam Wagstaff for much helpful advice; and George Lobell and Gale Epps at Prentice Hall for their excellent advice and guidance during the publication process. ..
Finally, and most important, I want to thank my wife Susan and children Debby, Daniel, and Jonathan for their patience and understanding while this book was being written.
Acknowledgments for the Second Edition
I would like to thank all those who took the time to send me corrections and suggestions that were invaluable in preparing this second edition, including Arthur Baragar, Aaron Bertram, Nigel Boston, David Boyd, Seth Braver, Michael Catalano-Johnson, L. Chang, Robin Chapman, Miguel Cordero, John Cremona, Jim Delany, Lisa Fastenberg, Nicholas Fiori, Fumiyasu Funami, Jim Funderburk, Andrew Granville, Rob Gross, Shamita Dutta Gupta, Tom Hagedorn, Ron Jacobowitz, Jerry S. Kelly, Hershy Kisilevsky, Hendrik Lenstra, Gordon S. Lessells, Ken Levasseur, Stephen Lichtenbaum, Nidia Lopez Jerry Metzger, Jukka Pihko, Carl Pomerance, Rachel Pries, Ken Ribet, John Robeson, David Rohrlich, Daniel Silverman, Alfred Tang, and Wenchao Zhou.
Acknowledgments for the Third Edition
I would like to thank Jiro Suzuki for his beautiful translation of my book into Japanese. I would also like to thank all those who took the time to send me corrections and suggestions that were invaluable in preparing this third edition, including Bill Adams, Autumn Alden, Robert Altshuler, Avner Ash, Joe Auslander, Dave Benoit, Jfirgen Bierbrauer, Andrew Clifford, Keith Conrad, Sarah DeGooyer, Amartya Kumar Dutta, Laurie Fanning, Benji Fisher, Joe Fisher, Jon Graft, Eric Gutman, Edward Hinson, Bruce Hugo, Ole Jensen, Peter Kahn, Avinash Kalra, Jerry Kelly, Yukio Kikuchi, Amartya Kumar, Andrew Lenard, Sufatrio Liu, Troy Madsen, Russ Mann, Gordon Mason, Farley Mawyer, Mike McConnell, Jerry Metzger, Steve Paik, Nicole Perez, Dinakar Ramakrishnan, Cecil Rousseau, Marc Roth, Ehud Schreiber, Tamina Stephenson, Jiro Suzuki, James Tanton, James Tong, Chris Towse, Roger Turton, Fernando Villegas, and Chung Yi.
Email and Electronic Resources
All the people listed above have helped me to correct numerous mistakes and to greatly refine the exposition, but no book is ever free from error or incapable of being improved. I would be delighted to receive comments, good or bad, and corrections from my readers. You can send mail to me at jhs@math. brown. edu
Additional material, including an errata sheet, links to interesting number theoretic sites, and downloadable versions of various computer exercises, are available on the Friendly Introduction to Number Theory Home Page:
www. math. brown. edu/~ jhs / frint. html
Joseph H. Silverman ...
