Why do people say that prime numbers are "random"? Compared to most of the people who frequent this place I suppose I am not very smart, but I do have a solid basic and somewhat intuitive understanding of mathematics.
Now prime numbers have always intrigued me, and I know for a fact that prime number must have a pattern. It is an unavoidable fact. I only wish I knew enough of higher mathematics in order to figure it out on my own and prove it once and for all. I dont though, so all I can do is state what I know and hope that someone will be kind and maybe give me more info or even one day solve the problem that has been driving my crazy for years and apparently so many other true mathematicians for long time.
What is the pattern?
Here is what I know: 1- Prime numbers are a fact a simple reality. 2- Because they always appear in the same places no matter what system you use to count them i.g. base 2 base 3 base... well any base don't matter it is all the same in the end. They must have a real pattern. 
So What is the pattern?
And more to the point, Why do so many people seem to think that they are random? 
 A: The primes are obviously not literally random, but viewing them as random is a surprisingly useful heuristic for coming up with plausible conjectures about the primes.  So while the primes are not random, in some ways their behavior resembles random things.  Terence Tao has some slides here that show how viewing primes as random seems to make accurate predictions of some of their properties.  (The discussion starts at slide 10.)
If you're interested in more details, search for the "Cramér model" of primes.
A: It might help if you forget about primes as such for a while, and try to decide what you mean by the word "pattern", or to put it another way, what you are prepared to accept as a pattern.  To illustrate what I mean here are three possible answers to your question.
(1) The primes obviously have a pattern, namely "the pattern of primes".  In other words, because they are an interesting set, that in itself means that they form a pattern.  This interpretation makes the question easy but not very interesting.
(2) "Having a pattern" means "being recognisable by a DFA".  In this case the primes do not form a pattern.
(3) "Having a pattern" means "being recognisable by a Turing machine".  In this case the primes do have a pattern.
Presumably what you want is somewhere in between (2) and (3).
