Primes: Randomness and Prime Twin Proof

Written by Martin Winer


Randomness and Prime Twin Proof

Martin C. Winer

Referring sites: I’m greatly appreciative of sites that have found my work interesting and have linked to me: Most Notably, I appreciate:

Google Directory Google Prime Directory DMOZ Open Directory Project DMOZ Open Directory H. Peter Aleff @ Recovered Science

Introduction Overview The purpose of this work is to look into some long pondered questions. First, isrepparttar distribution of primes acrossrepparttar 127645 number line random? Next, what is random anyways? Finallyrepparttar 127646 theories and axioms derived are used to solverepparttar 127647 long discussed “Prime Twin Problem” to show possible applications ofrepparttar 127648 understanding of what it means to be random.

Sieves and Patterns Consider all odd numbers starting at 3.

Mark a 1 onrepparttar 127649 number line whererepparttar 127650 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 thatrepparttar 127651 numbers corresponding torepparttar 127652 zeros between 3 and 3^2=9 are also prime (5 and 7).

3)The length ofrepparttar 127653 pattern is 3

Considerrepparttar 127654 pattern formed by 3 and 5:

1)repparttar 127655 pattern is 110100100101100...

2) Note thatrepparttar 127656 numbers that correspond to zeros between 5 and 5^2=25 are also prime (7,11,13,17,19,23)

3) The length ofrepparttar 127657 pattern is 3x5=15

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

If we let P(x) berepparttar 127658 xth prime starting from 3 such that

P(1)=3, P(2)=5, P(3)=7 and so on, we can considerrepparttar 127659 patterns on a larger scale.

Definition of Pat(n) Suppose we define a function Pat(n) which will producerepparttar 127660 string of ones and zeros as defined above from P(1) to P(n).

I.e. Pat(1) = 100…

Pat(2) = 110100100101100… (That’srepparttar 127661 pattern or sieve of 3 and 5)

In such a case,

(1) The pattern will consist of 1's and zeros corresponding torepparttar 127662 products and non-products ofrepparttar 127663 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 notrepparttar 127664 product of any prime below it. The first unique contribution a prime factor gives torepparttar 127665 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 inrepparttar 127666 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 ofrepparttar 127667 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. Inrepparttar 127668 case ofrepparttar 127669 prime number 3, we know that 3 is prime and that all (other) multiples of 3 are not prime. However, when we come torepparttar 127670 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 ofrepparttar 127671 prime number 3. Therefore,repparttar 127672 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 thatrepparttar 127673 odd multiples have no primes less than P(x) in their prime factorizations

Examples: Considerrepparttar 127674 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 torepparttar 127675 prime number 3.

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

Powers of a prime It turns out that powers of primes (greater thanrepparttar 127676 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 makerepparttar 127677 attempt. You can only say that it followsrepparttar 127678 patter 10… forrepparttar 127679 number of attempts you made because onrepparttar 127680 next attempt,repparttar 127681 pattern may change. Thus it is impossible to ever say that an infinite length pattern follows a certain pattern unless you are aware ofrepparttar 127682 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 berepparttar 127683 number of smallest repeating units inrepparttar 127684 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 Takerepparttar 127685 pattern:

Pat(2) = 110100100101100…

This repeats every 3rd and every 5th. Note it repeats every 9th as well but that’s notrepparttar 127686 smallest repeating unit becauserepparttar 127687 every 3rd subsumesrepparttar 127688 every 9th. Thusrepparttar 127689 mr of this pattern is 2.

Some Examples for Clarity So,

100000000… is no more or less random than

100…, or


(Because in all cases mr = 1)


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)

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

Cooling Down by Evaporating

Written by Thomas Yoon

Evaporator coils of air conditioning systems are sometimes called cooling coils. But cooling coils are also used in air-handling units. These contain chilled water. These are completely different from those used in direct expansion air cond units.

So cooling coils is not an accurate word to use in small air cond units. Evaporator coils should be used instead. Evaporator tubes must contain refrigerant liquid that can evaporate to become gas.

In an air conditioning system, whenrepparttar liquid refrigerant absorbs heat, it turns to gas. The heat is thus transfered torepparttar 127644 refrigerant. That's howrepparttar 127645 heat is moved from one location to another location.

The evaporator coils are located inrepparttar 127646 low-pressure system of a refrigeration circuit.

A word of caution!

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