Why do we consider prime numbers important, and what are their applications other than number theory in pure math? I know that Number theory is devoted to studying prime numbers, but there must be other reason to study the properties of prime numbers, right?
I would start by saying that much of number theory, to the extent that you can really describe number theory in a single sentence, is devoted to solving equations with integer solutions. And it turns out that understanding how these equations behave with respect to primes is often the key to understanding how they behave with respect to all integers.
Prime numbers are also important in understanding a great many concepts in abstract algebra, which generalize way beyond number theory. For example, if we look at an object with a finite number of symmetries, primes play key large role in understanding such symmetries (what I'm alluding to here is finite group theory).
Also, much of the abstract machinery developed to understand primes has found applications in other areas, particularly geometry. When we look at the set of zeros of a polynomial or complex analytic function in space, we do so by trying to understand the irreducible components of these sets, which correspond to something called "prime ideals."
I won't even get into the Riemann Hypothesis, except to say that a single conjecture (which we are unable to prove at present) is simultaneously a statement about primes, complex functions, convergence of series, and random matrices, to name just a few of the myriad formulations.
But perhaps the best reason to study primes is that they are simultaneously elementary and mysterious. It's remarkable just how little we know about these numbers after pondering them for millennia. For many mathematicians, that alone is sufficient motivation.
The additive structure of the integers is trivial: it's generated by 1. This is in essence the Peano axioms.
The multiplicative structure of the integers is not trivial: it's generated by prime numbers. In other words, prime numbers are the multiplicative building blocks of the integers in the sense that every nonzero number is either a prime or a product of primes (the empty product gives 1).
This gives a rich ring structure to the integers. Number theory is about that structure. Add to this the amazing regular irregularity of the primes and you get the richness of analytic methods.
One reason is that prime numbers are the basis of RSA cryptography, which is based entirely on using large prime numbers in clever ways. Studying the primes directly can change how secure we believe the RSA cryptographic algorithm to be. Currently we believe it to be very secure, but an unexpected advancement in the study of primes could lead to a way to break it, meaning we would have to change to something else.
This certainly has real-world application in that RSA cryptography is (to my knowledge) the standard of many important and sensitive information (think banks, credit cards online, etc).
Relative primes are also useful when building gear trains. It helps to reduce wear. If one gear has a number of teeth that is a factor of the other gear the same teeth always come into contact with one another.
For instance if one gear has 10 teeth and the other 20, the first tooth on the 10t gear always comes into contact with the 1st and 11th teeth on the 20t gear.
If, on the other hand, one gear has 11 teeth and the second has 20 teeth (20 is not prime but 11 and 20 have no common factors) the first rotation the first tooth contacts the 1st and 12th teeth on the first revolution and the 3rd and the 14th on the second, the 5th and 16th on the third, and it will take 21 revolutions to get back to the first tooth.
In nature prime numbers show up time to time. Cicadas live under ground for prime numbers of year before coming up and making lots of racket (here is a blog post about it http://www.factodiem.com/2010/04/prime-years-of-life.html)
Of course lots of thing are important to study, not because of what they HAVE been used for but for the things they WILL be used for.
Hope this helps.
The prime numbers have connections to pseudo random numbers. They might also have connections to "true randomness", but I'm not aware if there has been much progress on the conjectures which point in this direction. I wonder whether the Riemann Hypothesis is equivalent to a statement about the relation of the prime numbers to "true randomness", or whether it would at least imply such a statement.