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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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    ...or how primes are used in cryptography, RSA system. There are simply many things about primes in maths because the concept of a prime number is so important...they are the building blocks for our number system. So you see that it is too vague to just ask for stuff on primes, you could mean any one of these things and more!2012-06-10

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

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    See also http://math.stackexchange.com/questions/153022/a-good-reference-to-begin-analytic-number-theory.2012-06-04
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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.