Numbers, yes, but Prime Numbers are dependent on the idea that being divisible by only one and itself are important. There are infinite such ways to group numbers. We do Primes partly because they're useful, and also because they're cool.
"Random" is really a bit strong. They're certainly not very random in the Kolmogorov sense: the shortest program that generates the first N prime numbers is much, much shorter than constant*N. There are all sorts of obvious regularities (e.g., all are odd, all are 1 or 5 mod 6, etc.) and it's no good saying "well, they're random once you take those regularities into account" because taking those regularities into account uniquely determines the prime numbers. There are subtler deviations from the simplest random models, too; for instance, if you just pretend that each number n is (independently) prime with probability 1 / log n, you get what appears to be the wrong answer for the density of twin primes (pairs n,n+2 both of which are prime).
So no, "random as far as we currently know" is a pretty terrible description of the distribution of prime numbers.
We can prove the prime number theorem, we know roughly how many primes there are in a span of numbers and we know the approximate frequency of prime numbers in that span. We know on average how far apart the primes are in a particular span of numbers. All from the prime number theorem and related theorems. This stuff has been proven. Likewise, if the Riemann hypothesis is true then primes would have a very regular distribution.
What we can't yet do is create a generator function that catches most primes. We can create generators though (For example: (n! +1) is prime but it misses tons and tons of primes.) There are no hard fast rules that catch a majority of only primes. This probably doesn't qualify as "random distribution" unless you take a really simple definition of random. It's the non-randomness of their distribution that allows us to test the primality of numbers for cryptography.
I agree with the point you're making, but unfortunately almost all the particular statements above are wrong :-).
n!+1 is not always prime. For instance, 4!+1 = 25 = 5^2, 5!+1 = 121 = 11^2, 6!+1 = 721 = 7.103.
If RH is true, then the number of primes < x differs from the naive prediction you get from saying n is prime with probability 1/log(n) by at most something like sqrt(x) log(x). That's entirely comparable to the accuracy you'd get if the primes really were randomly distributed. (More precisely: the variance of something that's 1 with probability p, else 0, is p(1-p), which is roughly equal to p when p is small, and variances of independent things are additive, so on the random-distribution hypothesis the variance of pi(x) would be comparable to the sum of 1/log(n), which is to say n/log(n). I haven't actually done the calculations, but I think this means that with probability 1 some bound of the same type that you get from RH should hold. So the bound you get from RH doesn't show that the primes are any more regularly distributed than the simple random model suggests.
Similarly, "primes" generated according to the simple random model would, with probability 1, obey theorems of the sort that we know how to prove about the distribution of primes. (And some that we don't know how to prove.)
It is possible to define mathematical functions that recognize primes, or that take on all the primes (and nothing else) as values. But they're generally contrived and inefficient to evaluate.
Could you offer a citation? I tried to find the result you mention, and I'm at a loss to even pin down a definition of "random distribution" that might be used.
The closest I can come is the coincidence that a random variable is used in the proof of the Prime Number Theorem, but that's just a modeling technique.
At least I did find this awesome quotation, which seems to play both sides:
In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171).
The shortest program that generates the differences between the first N primes isn't much longer than the program that generates the first N primes (certainly not enough to break the constant*N barrier).
but Prime Numbers are dependent on the idea that being divisible by only one and itself are important
I confess that I only have an undergraduate degree in Mathematics, but with that caveat I have a very different understanding: importance is a value judgement, and concepts in number theory such as prime-ness hinge on mere observations about the properties of numbers. People may assign importance to the property, but the property exists independently of how people feel about it.
Prime Numbers are dependent on the idea that being divisible by only one and itself are important
Whether or not we care about a particular property does not affect whether or not some numbers exhibit that property -- they'd still be prime even if we didn't care about primality.
Your use of the word "random" seems to be a non sequitur. Could you explain that connection?