What are the essential number theory texts that every serious student of number theory should read?

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Undergrad or postgraduate? – Sanath K. Devalapurkar Jun 21 '14 at 21:38
@SanathDevalapurkar Both. I am an undergrad who is very interested in number theory but the subject is so vast, I'm not sure how to navigate it. – user159417 Jun 21 '14 at 21:40
I can't help you much, I'm afraid- I'm a category theorist. – Sanath K. Devalapurkar Jun 21 '14 at 21:42
You also have to understand that there is analytical number theory and algebraic number theory. They are of course connected. – Nicky Hekster Jun 21 '14 at 21:42
@SanathDevalapurkar: So how many papers have you written on category theory, "category theorist"? – user55315 Jun 21 '14 at 22:11

Have a look in Classical Introduction to Modern Number Theory by Kenneth Ireland, Michael Rosen and also the classic An Introduction to the Theory of Numbers, by G.H. Hardy & E.M. Wright. I can also recommend Introduction to Analytic Number Theory, by Tom Apostol.

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My recommendations would be:

• Hardy and Wright: An Introduction to the Theory of Numbers (old but still a great read)
• Stewart and Tall: Algebraic Number Theory (a very readable introduction to ANT)
• Alan Baker: A Concise Introduction to the Theory of Numbers (brilliantly written)
• Ireland and Rosen: Classical Introduction to Modern Number Theory

There are two books on analytic number theory by Apostol which are both also masterpieces.

But definitely avoid A. Weil's book "Basic Number Theory", unless and until you are much more advanced.

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Welcome back! – Mike Miller Jun 22 '14 at 3:02

I think that if you get another twenty answers, they will all mention Hardy & Wright. Your library might have it but you'll probably have to put in some kind of storage retrieval request.

I'm not sure if Old John's referring to this book:

• Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002)

That's a very good one. I also recommend:

• Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007) (Beware the long list of errata, though).
• H. Davenport, The Higher Arithmetic, 7th ed. 1999, Cambridge University Press
• Benjamin Fine & Gerhard Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Boston: Birkhäuser, 2007
• Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980)

It might also be a good idea to look at books devoted to the Fibonacci numbers.

As for books to avoid: books that look like they were typeset on a typewriter or on Microsoft WordPad and books that make strange claims about how knowing number theory is directly correlated to capital investment success.

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Books devoted to Fibonacci numbers (there are such things?) are not so important for a serious number theory student. – KCd Jun 22 '14 at 2:56
I dare you to leaf through Koshy's Fibonacci and Lucas Numbers with Applications and maintain that view. – Robert Soupe Jun 22 '14 at 3:02
After skimming a few chapters, my view has not changed. I am not doubting Fibonacci numbers are connected with some worthwhile topics in number theory (Euclid's algorithm, continued fractions, Pell's equation, algebraic integers), but Koshy's book is not written specifically for strong students in number theory, which is what the question being asked is about. – KCd Jun 22 '14 at 5:27
@KCd Well, alright. – Robert Soupe Jun 22 '14 at 17:03