What are the essential number theory texts that every serious student of number theory should read?
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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.
My recommendations would be:
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.
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:
That's a very good one. I also recommend:
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.