The sequence of primes is entirely deterministic. There are even nth prime formulae; however, the formulae are of exponential complexity...or worse.
Addendum: Oh, there's a link. The site explains how the sucession of primes is not so apparantly random after all.
It gets better than that.
There exist polynomials p(n) (n integer) that only take on prime or negative values. Even bettern there exist polynomials p(n) that can be every prime number and some negative values, but nothing else!
There even exists a formula for the n-th prime number! (I've only heard this referenced though, I have never actually seen it)
What if this is true, why are they still searching for big prime numbers with computers and stuff. Why don't they just plug 100,000,000,000,000,000,000,000,000,000,001 in for the n and get a massive prime number. I really have to doubt about this part.
What are the functions you were refering to that take on all prime numbers and some negative numbers. Those sound kind of interesting.
If T(f; n) is the time it takes to calculate f(n), the T(f; n + 1) is approximately T(f; n) times some constant k > 1. Attempting to calulate the 100,000,000,000,000,000,000,000,000,000,001st prime would probably require more eons than there are Angstroms in a lightyear.
See http://mathworld.wolfram.com/PrimeFormulas.html for some prime generation formulae, and http://mathworld.wolfram.com/PrimeNumber.html for general info. Mathworld is your friend. :)
No, lots of explicit formulae at http://mathworld.wolfram.com/PrimeFormulas.html
Separate names with a comma.