Primes:

Randomness and Prime Twin Proof

Martin C. Winer

martin_winer@hotmail.com

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Introduction
Overview
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 anyways? 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.

Sieves and Patterns
Consider all odd numbers starting at 3.

Mark a 1 on the number line where the number is a product of 3, (including 3×1), 0 otherwise. We get a pattern (sieve) such as:

1 0 0 1 0 0

3 5 7 9 11 13

1)The pattern is 100…

2)Note that the numbers corresponding to the zeros between 3 and 3^2=9 are also prime (5 and 7).

3)The length of the pattern is 3

Consider the pattern formed by 3 and 5:

1) the pattern is 110100100101100…

2) Note that the numbers that correspond to zeros between 5 and 5^2=25 are also prime (7,11,13,17,19,23)

3) The length of the pattern is 3×5=15

Definition of P(x) [The Xth Prime]
In general

If we let P(x) be the xth prime starting from 3 such that

P(1)=3, P(2)=5, P(3)=7 and so on, we can consider the patterns on a larger scale.

Definition of Pat(n)
Suppose we define a function Pat(n) which will produce the string of ones and zeros as defined above from P(1) to P(n).

I.e. Pat(1) = 100…

Pat(2) = 110100100101100… (That’s the pattern or sieve of 3 and 5)

In such a case,

(1) The pattern will consist of 1’s and zeros corresponding to the products and non-products of the n composing prime factors,

(2) The numbers corresponding to zeros between P(n) and P(n)^2 are guaranteed to be prime

By Haadi