Mostly they're good for creating reusable but nearly unbreakable crypto keys. The fact that prime numbers can only be divisible by themselves and 1 makes it possible to create numbers that are a) extremely large, and therefore part of a potentially extremely large number set, and b) can only be factored by a very limited set of other numbers, all but one of which need to be determined before the last factoral can be unlocked.

Take two primes and multiply them together. The resulting number can only be divided by those two primes, and of course 1 and itself. Not so difficult to figure out what those primes are with a number like, say, 77, but figuring out which two primes are multiplied together to create a 4096-digit number... that takes a lot more effort. All you can really determine from the start is that there must be 4096 digits total in the two prime keys, but even if you are also fairly certain those two numbers each have exactly 2048 digits, that's a lot of 2048-digit primes to try to divide your 4096-digit crypto key by, using the brute force method.

This kind of thing, of course, makes it easier to crack factoral encryption as well. We might end up having to move of to some other one-way crypto algorithm besides factorals soon.

Factorization. There's no such thing as a factoral, which as usual means I not only called it that not once, but twice.

I'm getting old.

Am I an Idiot or a genius ?, The primes are not random in anyway, shape or form. If one takes a random number then asks if it is prime the result appears to many to be effectively random. However this only occurs because of the selection technique. Primality is dependent on previous numbers, prime or non-prime.

Kind of. The main problem is nobody's ever been able to come up with a selection technique that's more accurate than the brute-force method of trying to divide an integer by all the integers less than the integral part of its square root. Again, not much of a problem for small primes, but a huge time-consuming problem when you go over 100 digits, even for supercomputers.

Every other method either will miss some primes, or worse, declare some integers prime that actually aren't, and nobody realizes they aren't until the crypto key based on them, which shouldn't be cracked even by a distributed net of all the computers in the world for at least 100 years, gets cracked in months by a hacker who decides to run the keys through some root detector the creators of the key overlooked, or in many cases hadn't even been invented yet.