For example 2 adjacent gaps cannot be equal if they aren't multiple of

6. For example the gap between 2 pairs of twins is at least 4. For

example each prime number has the form 2n+/-1, 3n+/-1, 4n+/-1, 6n+/-1.

Each pair of twins has the form 12n+-1, there are approximate formulas

for the nth prime and the number of primes < x and so on. You cannot

call all this random ore unpredictable. Of course the prime numbers are

distributed as regularly as possible, that's a tautology. In

mathematics everything is as regular as possible. Is pi random? Build

P=2,357111317192329
, and you have the same case as pi. Consider the

primes to be an irrational number, and there are no problems. If you

mean there is no formula f(n) which produces primes for each n, then

you are right. In this sense primes are random. (I am not quite sure

there is a formula p=[k^n^3] (H.W.Mills) which is said to produce only

prime numbers). If you define "formula" as an algorithm, as a

calculation instruction such as the sieve of Eratosthenes, then the

primes are not random but simply what they are. Perhaps the compound

numbers are random? Or are they only non-transparently complicated?

Werner