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I'm looking for a good introduction to Primes Numbers, their properties, and some of the better known theorems concerning them.

I would prefer references assume knowledge of undergraduate level real analysis.

Thanks to all in advance!

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It would help improve the quality of the recommendations if you provided some information on your mathematical background. This would ensure that the recommended books are not too simple or difficult for you. – Eugene Jun 4 '12 at 22:39
@Eugene I was thinking not placing any restrictions on the request may allow for a greater range; but, you're right, I'll specify a lower bound. – ThisIsNotAnId Jun 4 '12 at 23:12
An introduction to prime numbers shouldn't even involve real analysis! They are a basic mathematical creation coming from the integers. As for theorems/properties, there are millions spread over many different branches of need to tell us what questions about primes you would like to study/answer. – fretty Jun 9 '12 at 13:59
@fretty In the OP, I state the properties/theorems need to be "better known." Would this narrow it down to a particular set of number theory books? – ThisIsNotAnId Jun 10 '12 at 5:58
It doesn't tell me what kind of theorems you mean. Prime numbers are not a branch of maths but a type of number that behaves nicely. Your question is ambiguous, yet everyone else seems to have assumed you mean the prime number theorem in analytic number theory. Do you mean the classical number theory of what prime numbers are, how they provide unique factorisation and help to solve congruences. Or do you mean the algebraic number theory of how prime numbers split in number fields and how prime ideals are a generalisation. Or do you mean analytic number theory, PNT, Dirichlet's theorem...etc – fretty Jun 10 '12 at 7:25

Try The Prime Numbers and Their Distribution by Tenenbaum and Mendès France.

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See also…. – lhf Jun 4 '12 at 22:33

I liked Ribenboim, The Little Book of Big Primes. It has been updated. Or at Amazon

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I would recommend Apostol's Introduction to Analytic number theory.

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Even assuming undergraduate real analysis I think John Derbyshire's Prime Obsession is worth a look, and then, for its treatment of the Prime Number Theorem and generalizations, G.J.O. Jameson's The Prime Number Theorem. I recommend these as a non-expert who found them accessible but not patronizing.

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