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I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?

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This question is fine for MO if you can give more detail -- what year of undergraduate studies is this course? what is the syllabus? has this course been offered before? – Yemon Choi Sep 20 '10 at 20:01
Do you mean that the duration of the course is one month, or that in one month from now you will teach a course lasting a semester? – Jason Polak Sep 20 '10 at 20:06
Find out who taught the course before you and get his or her notes! – David Speyer Sep 20 '10 at 20:27
This question highlights the old saying: "A lecture is a system for transferring the lecturer's notes to the students' without going thru the minds of either.". Why not just give the students a set of photocopied notes and be done with it. The way most lectures are conducted is absurd and utterly wasteful of everyone's time. – user8232 Sep 21 '10 at 6:22
As a former student and present lecturer, remarks such as the previous one really get my goat :-( They also don't seem particularly relevant to the present question, although since the question is so sparing with information it's not clear what is relevant – Yemon Choi Sep 21 '10 at 10:42

Stein's book may be useful (and it is free):

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Boo, self promotion! Shame on you -- what's that? Free?? Cool. :) – Pete L. Clark Sep 21 '10 at 0:25
Now Ben seems to have hijacked William's answer. – Felipe Voloch Sep 21 '10 at 14:05
I'm teaching out of William's book right now and I can heartily recommend it. – JSE Sep 21 '10 at 17:51
I also think this is a very nice option. You should be able to read it fairly quickly to get ready for the course, and the students should enjoy the emphasis on real world computational examples. – Andrés E. Caicedo Sep 21 '10 at 18:35

I don't think the OP has provided enough information to get a useful answer to his/her precise question (what text to learn quickly from).

What level is the course being taught at? High school? Undergraduate for non-majors? Undergraduate for majors but without specific knowledge of any other undergraduate math courses beyond calculus? Undergraduate assuming some basic analysis and/or algebra? Graduate level? Something else??

As others have said, a perfectly reasonable thing to do when you are teaching any course for the first time and don't have strong opinions / too much expertise about it is to look at the textbook(s) that others have used who have taught the course recently. Thumb through them a little bit, then ask them how they liked the book and how well it worked for the course. If you found anything confusing or problematic in the book, ask them about that.

I think someone with a PhD in mathematics (for the sake of argument, I'll assume the OP has one) should be able to pick up and read a textbook for any undergraduate class within a month and then be able to teach the class with a reasonable amount of competence. Of course, real insight takes more time than that, and it is not reasonable to expect that someone conscripted into service with one month's worth of notice (why is this, exactly?) will be able to provide that.

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For an introductory undergrad course I'd say the book to use by a long-shot is Kenneth Rosen's Elementary Number Theory and its Applications

The theory is all there, but it's placed nicely in a context appropriate for a mixed bag of undergrad students by a large number of interesting-but-doable exercises and informative historical notes. Modern applications to computer science, cryptography, etc are all there and can be emphasized (or not) as you see fit.

This is what I'd read if I were you. Last time I checked, the book was annoyingly expensive - but this is the only criticism of it I have. Most students give this book very favorable reviews, too.

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Strange that a used bookseller will sell it for \$257 used when it is \$101 new. – Joseph O'Rourke Sep 21 '10 at 1:08
Try searching used on the first edition is available for a dollar, the third edition (1992) for ten dollars, fourth edition (2000) for thirty six dollars. – Will Jagy Sep 21 '10 at 4:04
I like Rosen's book, as well, but I suggested Dudley because it is very readable and you can get for $10 new from Dover. – Micah Milinovich Sep 21 '10 at 15:19
I second that recommendation. I took number theory with Rosen's book, and while it's not as theoretical as I would like, I learned a lot, and it's relatively reader friendly. – Daniel Miller Oct 8 '10 at 15:22

What about "The Little Book of Bigger Primes" by Ribenboim (see 1 for the Amazon link)? I personally think this is a great introduction to the field of number theory and I have enjoyed it very much a few years ago. It is clear and nicely written.

(Just to be clear: We are not talking about a course that also involves notable parts of algebraic number theory, are we?)

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If it is a course elementary number theory, look at "Elementary Number Theory" by Dudley.

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I think Jones + Jones

would be a good all-around introduction. It has solutions to every problem in the back, which can be helpful for self-study.

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