Introduction to number theory

by Joseph H. Silverman

《数论概论（原书第3版）》讲述了有关数论大量有趣的知识，以及数论的一般方法和应用，循序渐进地启发读者用数学方法思考问题，此外还介绍了目前数论研究的某些前沿课题。《数论概论（原书第3版）》采用轻松的写作风格，引领读者进入美妙的数论世界，不断激发读者的好奇心，并通过一些精心设计的练习来培养读者的探索精神与创新能力。《数论概论（原书第3版）》讲解清晰，语言生动，易于理解，适合作为高等院校相关专业学生的数论入门书，也可以作为有志于学习数论的读者的自学读物。《数论概论（原书第3版）》面向非数学专业学生，讲述了有关数论的知识，教给他们如何用数学方法思考问题，同时介绍了目前数沦研究的某些前沿课题。对于定理的证明，则强调证明方法而不仅仅是得到特定的结果。引言欧几里得独自迈入美妙的数论世界，尽管只有一次而且很快远去，但他的足迹还是深深地影响着后人。——米雷（Millay），1993自然数1，2，3，4，5，6，的起源已随着时间的消逝被人遗忘。我们不知道谁首先意识到像“三”这样的抽象概念，它可指三块岩石、三颗星、三个人等。译者序中文版序前言引言第1章什么是数论第2章勾股数组第3章勾股数组与单位圆第4章高次幂之和与费马大定理第5章整除性与最大公因数第6章线性方程与最大公因数第7章因数分解与算术基本定理第8章同余式第9章同余式、幂与费马小定理第10章同余式、幂与欧拉公式第11章欧拉必函数与中国剩余定理第12章素数第13章素数计数第14章梅森素数第15章梅森素数与完全数第16章幂模m与逐次平方法第17章计算模m的k次根第18章幂、根与不可破密码第19章素性测试与卡米歇尔数第20章欧拉φ函数与因数和第21章幂模p与原根第22章原根与指标第23章模p平方剩余第24章－1是模p平方剩余吗？2呢第25章二次互反律第26章哪些素数可表成两个平方数之和第27章哪些数能表成两个平方数之和第28章方程X4＋Y4＝Z4第29章再论三角平方数第30章佩尔方程第31章丢番图逼近第32章丢番图逼近与佩尔方程第33章数论与虚数第34章高斯整数与唯一因子分解第35章无理数与超越数第36章二项式系数与帕斯卡三角形第37章斐波那契兔子问题与线性递归序列第38章Ο，多美的一个函数第39章连分数的混乱世界第40章连分数、平方根与佩尔方程第41章生成函数第42章幂和第43章三次曲线与椭圆曲线第44章有少量有理点的椭圆曲线第45章椭圆曲线上模p的点第46章模p的挠点系与不好的素数第47章亏量界与模性模式第48章椭圆曲线与费马大定理附录A小合数的分解附录B6000以下的素数表参考文

Genres: Mathematics, Nonfiction, Textbooks, Number Theory, Science

Paperback

First published November 1, 2013

Reviews

[Mark Moon · Rating 5 out of 5]

A very nice introduction! It's as friendly as the name implies, and does get into some relatively advanced material towards the end, too.

[Hannah Thomas · Rating 4 out of 5]

This is really more of a computational textbook for a college class, but there are some areas that you have to use "proofs." To understand Number Theory, you really do need to some basic Abstract Algebra background. There are times in here that you do get that "background" knowledge of the abstract part, but it's very vague because only the "uniqueness" part is provided. It's the "existence" you need to understand and prove (that's where your background knowledge comes in handy!). The author really does want to work your mind around proofs and computation and that's the fun part :) If you really want to pursue further into mathematics, in the Abstract Algebra branch, this is a book for you!

[Chris · Rating 4 out of 5]

It lives up to the title. If you think mathematics is work and not play, you might find yourself surprised with this slim volume.

In addition to being an instrument of a peculiar kind of pleasure, it's free of data pseudoscience.



Programmers who want to learn more about the elliptic curves in cryptography could do worse than giving Joe Silverman 1/4th the price of a typical language/database doorstop.

[Shaun Zhang · Rating 4 out of 5]

It is a readable book. As the title shows, it is friendly. And all basic important theorems are well presented by examples and rigorous proofs. Recommend it for whoever first touches number theory.

[Xiangnong · Rating 4 out of 5]

As a high school student and a math competition participant, I found this book to be a very good enlightenment of my Number Theory knowledge. It is not extremely hard to understand. Some of them are even easy. Everything is introduced as a level which is just enough to be understood and applied.
Things I found interesting and helpful are those exercising examples provided. It will always provide a complete example to show an algorithm or explain how to prove a theorem. As an OIer once, I learned several algorithms including EGCD and Fast Powering several years ago, but I have neither tried to work it out by hand, nor tried to test and prove the validity of those algorithms.
One thing I learned from this book is that you need to test out and play around with numbers in order to discover some new patterns and modes. And I believe I will read this book again because this time I didn't really tried to work on those exercises. It's required, just like people need to go underwater in order to learn swimming.
Then I realized that, the key of this book is not to learn how to prove fancy theorems and how to become a master in Number Theory, but how to start working on studying numbers.
Because of that, this book purposefully serves as an introduction, not a guidance, it can only provide basic knowledges of Number Theory and some fundamental proofs. The solutions provided for practice problems is really brief and not very helpful. I personally found it more helpful to re-read the examples than reading the solutions.
I think this book is very helpful to students who know nothing about Number Theory and want to know about it and are not college students yet. College number theory definitely requires more skills and deeper understanding beyond this book, and college students can find more helpful books such as Elementary Number Theory and Its Applications.
But for students like me or even younger, it is a good way to let them know about what is Number Theory and what do mathematicians do with it.

[Poonam · Rating 5 out of 5]

nice introductory text. kinda weak on elliptic curves

[Robert Allen · Rating 1 out of 5]

As the expositions are minimal, as most of the problems are out of left field, they cannot be worked, much less understood without the answers which I was able to purchase on line for $30.00--and even the answers are difficult to understand!!!!! In short, this book is everything but friendly and Silverman and those on this thread who gave it a top rating are no more than frauds, cheats and liars.

[Hülya · Rating 5 out of 5]

bu kitabı da burda gördüm ya daha ne olsun :D silverman amcayada teşekkürler süper eğlenceli bir kitap yazmış. number theoryi hiç böyle bilmezdim ben :P

[Samantha · Rating 3 out of 5]

I read this book for Number Theory this past semester. It makes the concepts very accessible and was very enjoyable.

[Dev]

good

[Joe Cole · Rating 5 out of 5]

I took a college course in number theory and this was the text. I have found it to be an easy read, only requiring a few visits to Wolfram Alpha to clarify some background knowledge.