Uses for Prime Numbers

Aside from keeping Mathematicians guessing the only significant application of prime numers and prime number theory is encryption.

Encryption

Prime numbers are an integral part of modern internet security. The most common form of internet security is RSA security which has at its heart 2 very large prime numbers.

RSA security revolves around finding two large (2048 bit or 617 digit) prime numbers and multiplying them together to get your own, personal "modulus," which will be another very large number. You post your modulus for the world to see and if anybody wants to encrypt a secret number to send to you, all he has to do is cube it (multiply it by itself twice), divide by your modulus, and send you the remainder from the division.

Knowing the prime factors of the modulus, you can recompute the original secret number from the remainder that your friend sends you (the details are slightly complicated); but an enemy who intercepts the remainder -- even an enemy who knows your modulus -- in general cannot recompute the original secret number from it. The "secret number" can, of course, be some kind of letters-to-numbers translation of a textual message. (Warning: for this encryption to be strong, the secret number must be big enough that its cube greatly exceeds the modulus. In practical applications, a large number is added to the secret number before cubing.)

As far as anybody knows, there is in general no way to break this cipher faster than by finding the two prime numbers that were multiplied together to make your modulus. Consequently, cryptologists are very interested in how big a number of this sort can be factored, since that tells them how big they have to make their moduli. 