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This question already has an answer here:

What practical use are prime numbers? Why do we emphasise the teaching of prime numbers?

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marked as duplicate by William, Raskolnikov, Start wearing purple, user1337, Dan Rust Sep 3 '13 at 10:31

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    $\begingroup$ Your question isn’t really very clear. What context do you have in mind when you say that we emphasize the teaching of prime numbers? $\endgroup$ – Brian M. Scott Sep 3 '13 at 9:31
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    $\begingroup$ Cryptography is the only "practical use" that comes to mind. $\endgroup$ – daniel Sep 3 '13 at 9:46
  • $\begingroup$ To continue Brian's question (into possibly a different direction what he had in mind): at what level are you being taught about prime numbers? Knowing that helps answering your question. $\endgroup$ – Jyrki Lahtonen Sep 3 '13 at 9:51
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For real life applications see Real-world applications of prime numbers?. Why are prime numbers so important that we teach about prime numbers ? This question is not so easy as it seems. Of course, for elementary number theory, prime numbers are like the "atoms", and several questions involve prime numbers.
For mathematics in general, the value of prime numbers lies much deeper. For example, the distribution of prime numbers encodes very deep mathematical information in general (not only via the Riemann Hypothesis). Completions of the rational numbers naturally lead to $p$-adic fields, and the idea of being "prime" applies to many other structures (like prime ideals, prime geodesics etc.). So we emphasise teaching prime numbers because they lie at the very heart of mathematics.

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Primes are of prime importance in cryptography (take for example the RSA cryptosystem). See http://en.wikipedia.org/wiki/RSA_(algorithm)

However one does Mathematics simply for doing Mathematics. What Mathematicians think today, may find its application even more than 200 years later. Although many branches (like variational calculus) came up from practical issues, not every branch of mathematics can be even remotely related to some real-life problem.

Prime Number Theory is a very interesting topic. It not only consists of some exciting results and findings over the ages but it actually enhances our mathematical thinking and imagination.

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  • $\begingroup$ > one does Mathematics simply for doing Mathematics -- that's fabulous if you can get some generous person to pay you to work on stuff that (maybe) has some unintended benefit 200 years from now. That kind of generosity is getting increasingly rare, though. Presumably that's how Yitang Zhang ended up working at Subway. $\endgroup$ – bubba Sep 3 '13 at 10:40

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