A random process or event is one where for every suspected (macro or micro) pattern (other than the pattern of the event or process itself), there exists a change such that the suspected pattern no longer holds.

Corollary:

A random process or event is one with an infinite supply of complexity (neither redundant nor reducible changes).

Macro/Micro Pattern

Take for example the pattern

1010111110101111

****^^^^****^^^^

For every region *= 1010 there exists a change such (to ^=1111) such that the suspected pattern never holds. HOWEVER, there a macro pattern of *^*^ (1010, then 1111, then 1010, then 1111 and so on).

Overall Purpose

The purpose of this work is to look into some long pondered questions. First, is the distribution of primes across the number line random? Next, what is random anyway? Finally the theories and axioms derived are used to solve the long discussed “Prime Twin Problem” to show possible applications of the understanding of what it means to be random.

Summary

Pat(n) is the fundamental building block by which composite and prime numbers are laid down along the number line. Pat(n) is a recursive algorithm which merges (algorithm described below) in the pattern 1 followed by (P(n)-1) 0’s with Pat(n-1).

Whenever a 0 occurs in Pat(n) between P(n) and P(n)^2 a prime occurs. Whenever a 00 occurs similarly, a prime twin occurs.

The probability of either a 0 or a 00 between Pat(n) (in general) approaches zero, but never reaches it.

Even though there are relatively small numbers of 0 and 00's (single and prime twin candidates respectively) in Pat(N) with large N, you can't rule out their existing between P(n) and P(n)^2 by virtue of the fact that Pat(n)'s complexity grows without bound with increasing N. That is Pat(n) grows as random as you care to make it with increasing N. (Random subsumes randomly distributed.)

Hence, there is a non-zero probability (although decreasing) that there will be a 0 or 00 for in Pat(n) between P(n) and P(n)^2 for for any large N.

Any non-zero probability (even a decreasing one) event given enough chances will eventually occur. Since we can take as many chances as we want (the number line is of infinite size), we will eventually get another prime twin or prime singleton. Thus the set of prime twins (and primes) is infinite because for any prime twin (or prime singleton) we can get the next one.

Even though the set of primes and prime twins is infinite, they are also effectively finite. That is, for increasing n, they become so sparse, so distant between instances that they are effectively finite, however strictly infinite.

All these observations are true for any allowable constellation of primes. That is any allowable constellation of primes will likewise be infinite, however, will ‘fizzle’ to near finiteness.

Summary 2

Pat(n) is random across Pat(n)'s for increasing n relative to the floor(P(n)/2)’th position.

"WHAT?!"

Let's break this one down:

What is P(n)?

P(n) is the nth prime starting at 3. P(1) =3, P(2) = 5, and so on.

What is Pat(n)?

Pat(1) = 100… (… means repeat everything to the left over and over again, hence 100100100 and so on)

Pat(2) =

P(1) 100100100100100

AND 010000100001000 ß this is 1 and P(2)-1 0’s shifted to

align the first 1 with the first

0 in Pat(1)

= 110100100101100 ß length = P(2)*P(1) = P