# Good Number theory book?

pivoxa15
I am doing a course called 'Number Theory'. It is an introductory course to the subject and some if not most of it is based on the book

'Number theory with computer applications'
by Ramanjuachary Kumandari and Christina Romero (1998).

Anyone have any experince with this book? If so do you recommand that I purchase it to include in my own personal library for future reference?

Thanks

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if you want a number theory book try hard to get hold of LeVeque's Fundamentals of Number Theory (though it isn't about computation; if you don't tell us what the syllabus is then there is no way we can suggest any alternate titles).

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I like Burton's book for an intro course on Number Theory. There's no computational stuff in it though.

gazzo
I second Matt, and bought LeVeque based on his advice - although it is dense and a tough read if you've never visited number theory before (maybe an analysis course would be helpful?)

pivoxa15
Subject Description:

This subject introduces the elementary concepts of divisibility; the basic theory and use of congruences; the properties of powers of elements in congruences, particularly Euler's theorem; the law of quadratic reciprocity; and basic properties of continued fractions and some applications. It develops applications of all of the above to primality testing, factorisation algorithms and cryptanalysis. Students should develop the ability to perform the algorithms inherent in the subject material; and to understand and present proofs related to the subject material. This subject demonstrates the extent and uses of elementary number theory, its applicability in other parts of mathematics, and its potential for application outside of mathematics.

Topics include factorisation, primes and greatest common divisors; congruences; primitive roots; quadratic reciprocity; continued fractions and Pell's equation; compositeness testing and factorisation; and applications to cryptanalysis.

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Then LeVeque would be good for most of it (and it is a nice read, honest, I wouldn't consider it tough until the later half which is of no interest to what you're doing). Computationally I don't know what to suggest since I suspect the only book I know on this is too far advanced for an introductory course, though you might want to look at it after you've taken the course to see what you need to do later, it is Cox's Primes of the Form x^2+ny^2, but it really is well beyond a basic course.

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Just a question, matt grime:
I'm sort of interested of finding a good book in number theory which is broad in scope, yet goes deep enough into each subject so one could actually learn something from it.

I have this idea that Hardy wrote something like this, but:
a) is that true?
b) is it any good?
c) is it way too "outdated" (dumb expression, my bad..)?
d) Are there other books beside the one by LeVeque I should look into?

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ramsey2879
since crytography is mentioned twice in the course description a book I know of may be of interest Richard Mollin "Fundamental Number Theory with Applications" 1998 edition. I am not able to recommend it through since I am not well versed in number theory or crytography. However, it has references to crytography throughout as well as a new interesting section on the subject. I brought the book since it had answers to all odd numbered problems.

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TenaliRaman
I would suggest,
Introduction to Theory of Numbers by Niven and Zuckerman
I know its a bit old (i wonder if its still available, except maybe in some library). But its definitely a nice book to learn number theory from.

I also come to like David M Burton and Hardy and Littlewoods books.

-- AI
P.S -> If you are looking at number theory from a computational perspective then have a look at this,
http://www.shoup.net/ntb/

One more book that's pretty comprehensive but hard to understand is,
Modern Computer Algebra by Gathen and Gerhard

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TenaliRaman said:
I would suggest,
Introduction to Theory of Numbers by Niven and Zuckerman
I know its a bit old (i wonder if its still available, except maybe in some library). But its definitely a nice book to learn number theory from.

I also come to like David M Burton and Hardy and Littlewoods books.

-- AI
Thank you, TenaliRaman!

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Hardy's book on Introductory Number Theory is at a slightly higher level than Burton's book. I find it organized a little weirdly, and would not recommend it as a sole reference for a first time venturer into number theory. I have not read LeVeque

For "applied" number theory I recommend : "The Mathematics of Ciphers - Number Theory and RSA Cryptography" by S.C. Coutinho

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TenaliRaman said:
I would suggest,
Introduction to Theory of Numbers by Niven and Zuckerman
I know its a bit old (i wonder if its still available, except maybe in some library). But its definitely a nice book to learn number theory from.

There's a '91 edition with Montgomery. It's an excellent text.

Hardy and Wright's book is a classic, and definitely organized differently from your standard text as Golkul mentioned. There was an edition in '78 that was updated by Wright with some new material.

The OP's course seems heavy on algorithms? A nice reference is "Handbook of Applied Cryptography", Menzies, van Oorschot, and Vanstone http://www.cacr.math.uwaterloo.ca/hac/

Also online is Mosers http://www.trillia.com/moser-number.html not for algorithms but a good variety of topics (though it's a short book) at a decent depth, plus it's free.

At a lower level than most books mentioned, but living up to it's title is "A Friendly Introduction to Number Theory" by Silverman. It's recent so it includes basic primality testing and crypto applications. Basic, but that's what you expect with a goal of "Friendly" and a book aimed for an intro undergrad class.

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shmoe said:
There's a '91 edition with Montgomery. It's an excellent text.

Hardy and Wright's book is a classic, and definitely organized differently from your standard text as Golkul mentioned. There was an edition in '78 that was updated by Wright with some new material.

The OP's course seems heavy on algorithms? A nice reference is "Handbook of Applied Cryptography", Menzies, van Oorschot, and Vanstone http://www.cacr.math.uwaterloo.ca/hac/

Also online is Mosers http://www.trillia.com/moser-number.html not for algorithms but a good variety of topics (though it's a short book) at a decent depth, plus it's free.

At a lower level than most books mentioned, but living up to it's title is "A Friendly Introduction to Number Theory" by Silverman. It's recent so it includes basic primality testing and crypto applications. Basic, but that's what you expect with a goal of "Friendly" and a book aimed for an intro undergrad class.
Thank you, shmoe!

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i must confess my attachment to leveque is that his is the book i read first, but more importantly i still read it.

it defines group, ring, equivalence relation, discusses primes and their distributions, proves things about primitivce roots modulo p^n and quadratic residues. in short everything that every mathematician ought to know about number theory. Baker is good and cheap too. Never read hardy.

neurocomp2003
Neal Koblitz: Number THeory and Cryptography (very technical)
was the book a prof suggested to me very interesting read but a bit high for my level when i was an undegrad...now I'm waiting to get the book when i have funds. If your looking for more computational books...mmm look under cryptography ...there's quite a few books out there and unfortunately the one i have is an ebook without a title so i can't give you the actual book reference, but there is a book called cryptography in C++.

TenaliRaman
By the way i forgot to add that i am doing a course in Computational Number Theory and Algebra in my first semester of masters. We are doing it from several books, we are mainly following,
1. Modern Computer Algebra by Gathen and Gerhard
2. The book by Victor Shoup (that i linked to in my earlier post)
3. Introduction to finite fields and their applications by Lidl and Niederreiter

Personally, i am also following,
1. Introduction to Theory of Numbers by Niven and Zuckerman
2. Applied Abstract Algebra by Lidl and Pilz

Just thought this info might help the OP.

-- AI

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matt grime said:
i must confess my attachment to leveque is that his is the book i read first, but more importantly i still read it.

it defines group, ring, equivalence relation, discusses primes and their distributions, proves things about primitivce roots modulo p^n and quadratic residues. in short everything that every mathematician ought to know about number theory. Baker is good and cheap too. Never read hardy.
Thank you, matt grime.

neurocomp2003
have any of you read koblitz? what are your opinions on the book if you have?

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neurocomp2003 said:
have any of you read koblitz? what are your opinions on the book if you have?
I'm a total novice here, neurocomp2003, so no, I haven't read koblitz.
Thank you for your suggestions, though.

If you want something informal and a little bit historic, try "Number: The Language of Science" by Tobias Dantzig.

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Icebreaker said:
If you want something informal and a little bit historic, try "Number: The Language of Science" by Tobias Dantzig.
Okay, I can't slight you now either, can I?
Thank you, Icebreaker..

I don't get it

THE CONFUSION

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Icebreaker said:
I don't get it

THE CONFUSION
My thanks was heartfelt; but I started thanking everyone who came with a suggestion for a book on number theory, and now, I cannot stop doing that.
Then, the next helpful person would feel slighted because I didn't thank him, but all the others..

If only you can thank by induction. :rofl:

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Icebreaker said:
If only you can thank by induction. :rofl:
:rofl: :rofl:

Eratosthenes
An easy to read book is Elementary Introduction to Number Theory by Calvin Long. It's a small book at only 124 pages. My version was printed in 1965 but there are newer ones available online. I've only read the first 3 of the 6 chapters but I think it is really good so far. This book is really good for someone who wants to learn on their own.

If you are just looking for something as a reference then some of the other books mentioned might be better since they cover more topics.

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Gokul43201 said:
I like Burton's book for an intro course on Number Theory. There's no computational stuff in it though.
i have it too (in a hebrew version), it's very nice, it has a nice historical coverage that doesn't harm the reading experience.
the book is called "Elementary Number Theory".