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 distribution of primes across number line random? Next, what is random anyways? Finally theories and axioms derived are used to solve long discussed “Prime Twin Problem” to show possible applications of understanding of what it means to be random.

Sieves and Patterns Consider all odd numbers starting at 3.

Mark a 1 on number line where number is a product of 3, (including 3x1), 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 numbers corresponding to zeros between 3 and 3^2=9 are also prime (5 and 7).

3)The length of pattern is 3

Consider pattern formed by 3 and 5:

1) pattern is 110100100101100...

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

3) The length of pattern is 3x5=15

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

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

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

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

I.e. Pat(1) = 100…

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

In such a case,

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

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

[Why? because a number is either a prime or a product of primes. A zero means that it's not product of any prime below it. The first unique contribution a prime factor gives to number line occurs at P(n)^2 = P(n)xP(n) because below that at say P(n)xP(n-1) can be rewritten as P(n-1)xP(n) and thus is already accounted for in number line. Thus a zero between P(n) and P(n)^2 is not a product of primes and must therefore be prime.]

(3) The length of pattern will be:

P(n) x P(n-1) x P(n-2) x ... x P(1)

Unique Contributions of P(x) Description As we build iteratively build Pat(n)’s over time, each successive prime adds to our knowledge of which numbers are prime and which numbers are not. In case of prime number 3, we know that 3 is prime and that all (other) multiples of 3 are not prime. However, when we come to prime number 5, 5 does not ADD to our knowledge that all multiples of 5 are not prime. For example 15 is a multiple of 5, but we already knew that this number was not prime because of prime number 3. Therefore, unique contribution to our knowledge as we build Pat(n)’s that a given prime (P(x)) provides us is given below:

Definition uniqueContribution(P(x)) For any prime, P(x), define

UniqueContribution(P(x)) = {P(x)*k; k is odd, k>=P(x), primeFactorization(P(x) contains no primes < P(x)}

In English… The uniqueContribution a prime number (P(x)) gives us as to which numbers are not prime while building successive Pat(n)’s is a function of all odd multiples of P(x) such that odd multiples have no primes less than P(x) in their prime factorizations

Examples: Consider prime 5.

5*5 = 25 is a unique contribution of 5

5* 15 = 5*3*5 = 75 is NOT a unique contribution because it has 3 in its prime factorization. Ie, we already knew that 75 was not prime thanks to prime number 3.

5*5*5 = 125 is a unique contribution.

Powers of a prime It turns out that powers of primes (greater than first power) are unique contributions.

Important Notes on uniqueContribution(P(x)) For larger P(x), uniqueContribution(P(x)) becomes increasingly difficult to calculate and more complicated. The unique contribution becomes more random as P(x) increases.

General Notes on Randomness Axioms of Randomness 1) All truly random patterns must be infinite length

2) A pattern is said to be random if there is an infinite supply of complexity

Black Box Pattern Paradox It can only ever be said that an infinite length pattern follows a pattern for a certain finite length. Suppose you have a machine that spits out 1’s and 0’s and it spits out 1010101010… for a certain number of times you make attempt. You can only say that it follows patter 10… for number of attempts you made because on next attempt, pattern may change. Thus it is impossible to ever say that an infinite length pattern follows a certain pattern unless you are aware of algorithm that generates it.

On Randomness of Primes Measure of Randomness in a Binary Pattern Let’s define a measure of randomness (mr) for a binary pattern to be number of smallest repeating units in lowest reducible pattern.

Definition of Lowest Reducibility: A pattern is reducible if it can be rewritten in a simpler, shorter form, such as:

11111111… is reducible to 1…

10101010101010… is reducible to 10…

Definition of Smallest Repeating Units Take pattern:

Pat(2) = 110100100101100…

This repeats every 3rd and every 5th. Note it repeats every 9th as well but that’s not smallest repeating unit because every 3rd subsumes every 9th. Thus mr of this pattern is 2.

Some Examples for Clarity So,

100000000… is no more or less random than

100…, or

100000000000000000000000…

(Because in all cases mr = 1)

However,

110100100101100… (every 3rd and 5th) is more random than those above since mr=2.

Some other interesting examples for clarity:

110… has mr = 2 because it has two repeating units of size 3, (the second offset by 1)