Computing Large Prime Numbers...Useless?

I used to think so. In fact, I believe it actually was useless, at least up until the 1970s. But since then, the properties of prime numbers have become quite useful for certain applications. One particular use of this computation is RSA encryption, which was briefly covered in a few of the courses I've taken here at UT.

I won't get into the specific mathematics of it all since Wikipedia/Google can provide a much better explanation than me. The basic idea is that it is easy to multiply two prime numbers together, but it is difficult to factor that product back into the original set of prime numbers. This is an example of a one-way function (also known as a trapdoor function), which basically means that the function is easy to compute but "hard to invert." Of course, given a small set of prime numbers, this isn't too difficult, despite the definition.

And that is why it is important to come up with large prime numbers. The larger the prime numbers, the more difficult it will be to factor the product back into the original pair of prime numbers. For example, could you easily factor 26,636,627 into two prime numbers? Certainly wouldn't be easy to do it with pencil and paper. Perhaps a computer could compute it if given enough time to run, but it might take a little while (certainly slower than factoring 6 for example).

Anyway, thought this was an interesting aspect to prime numbers. Oh and by the way, the factors are 3449 and 7723.