Books on the introductory number theory?

[bacte2013]

Dear Physics Forum advisers,

I am currently looking for an introductory textbook that covers the number theory without being too focused on the algebraic and analytical aspects of NT. My current underaduate research in the theoretical computer science and the Putnam preparation led me to the fascinating world of number theory. Basically, I am looking for one I can enrich my interest to the number theory, prepare for Putnam Competition, and aid my undergraduate research. I personally heard of the number theory books by Burton and Niven/Zuckerman/Montgomery, but I am not sure which will fit my purpose well. Could you recommend me some good books on the introductory number theory?

My research adviser personally learned from the algebraic and analytic number theory textbooks, so he could not help me out since the books he read require a strong knowledge in the abstract algebra and complex analysis. I am currently studying the real analysis (Rudin, Apostol), abstract linear algebra (Hoffman/Kunze), and abstract algebra (Artin, little bits per day).

 

[verty]

I had recommend some books here but I'm not confident that they would be all that great or better than any other in particular. There is a free book by Victor Shoup available that has some connection to computer science. You might like to look at that one.

 

[mathwonk]

there are a lot of good books on number theory out there. in my opinion some of the most famous tend to be a bit too brief, such as ones by Niven, although he is a modern master. If you want to challenge yourself I would suggest Gauss rather than Niven, Disquisitiones Mathematicae. I have always liked my copy of Introduction to Number theory by Trygve Nagell, but it is a little challenging as well. A fine number theorist friend of mine recommended the little book by Van den Eynden for bneginners, but that book has skyrocketed in price to become absurdly over priced. If you just want a nice little intro that is really aimed at young beginners, I think my son started out on the book by Underwood Dudley, which is also very cheap: less than $10 on amazon:

https://www.amazon.com/dp/048646931X/?tag=pfamazon01-20

 

[bacte2013]

there are a lot of good books on number theory out there. in my opinion some of the most famous tend to be a bit too brief, such as ones by Niven, although he is a modern master. If you want to challenge yourself I would suggest Gauss rather than Niven, Disquisitiones Mathematicae. I have always liked my copy of Introduction to Number theory by Trygve Nagell, but it is a little challenging as well. A fine number theorist friend of mine recommended the little book by Van den Eynden for bneginners, but that book has skyrocketed in price to become absurdly over priced. If you just want a nice little intro that is really aimed at young beginners, I think my son started out on the book by Underwood Dudley, which is also very cheap: less than $10 on amazon:

https://www.amazon.com/dp/048646931X/?tag=pfamazon01-20

Thank you very much for the detailed recommendation. I am not so sure if I am ready for Gauss (is it the same person that I know as one of greatest mathematician of all times?)...I narrowed my choice down to Niven, Nagell, Eynden, and Shoup (recommended by Mr. Verty). I went through sample chapters of Burton and Dudley, but their exposition are not my taste; I always like the books that present definitions, theorems, proofs, and corollaries in a categorical and flowing manner. I am currently exploring the Rocky Mountains, so I do not have any mean of accessing my university's college library. If you could recommend one from my list, which one is your choice (coverage of materials in depth and width, challenging problems, insights, etc.)!

 

[verty]

If you could recommend one from my list, which one is your choice (coverage of materials in depth and width, challenging problems, insights, etc.)!

I've read a few number theory books. None that I've seen have been terribly good. If I was choosing a book for myself, it'd be Ireland & Rosen. That's the one I'd pick for depth, challenge, etc.

I think you are in for a shock if you ever go to grad school, it's going to be a lot different. Better to get used to it now and use the proper books.

PS. So for example, if Mathwonk recommends Gauss, I would definitely read that.

 

[mathwonk]

well since you ask for one from your list, and i really want to you to make your own choice, i will suggest the book by shoup, suggested by verty. my reasons are that it looks good from the table of contents, and it is free, so if you don't like it you are not out any money. it is also more modern than my suggestions so incloudes topics not around in earlier days, like probabilistic primality testing, which has applications presumably today to code breaking and security.

http://www.shoup.net/ntb/ntb-v2.pdf

bear in mind all books will begin with the same standard topics: greatest common divisors, euclidean algorithm, and congruence arothmetic. shoup continues by introducing various topics from algebra like abelian groups and lineat algebra, instead of assuming them. which you say you still need to learn, another plus,