A Distribution Analysis formula.

[qpwimblik]

Using this formula the prime numbers should distribute themselves in a seemly chaotic balancing act between the negative and positive axis on your graph.
like this.

http://www.pyramidus.org/Images/DistributionFunction.jpg

And the formula.

1/2*((-1+abs(prime(n))+abs(prime(1+n)))*(abs(prime(n))+abs(prime(1+n)))-(-1+abs(prime(2+n))+abs(prime(3+n)))*(abs(prime(2+n))+abs(prime(3+n)))-(-1+abs(prime(4+n))+abs(prime(5+n)))*(abs(prime(4+n))+abs(prime(5+n)))+(-1+abs(prime(6+n))+abs(prime(7+n)))*(abs(prime(6+n))+abs(prime(7+n)))-2*prime(1+n)+2*prime(3+n)+2*prime(5+n)-2*prime(7+n))

 

[qpwimblik]

I wonder if you could see if a number is a prime or not using analysis of numbers put into this formula while trying to make a sequence of primes. So by finding a sequence of primes you can see if your number is a part of the sequence. It probably wouldn't be a fast primality test for small numbers but who cares about small numbers say below 200000000. One thing I can see straight away is that there is nearly the same amount of primes below 0 as above 0 and From tests I have found that if one of the primes is even out by one nth prime the results are distorted and the graph becomes unbalanced. Feel free to play with the function yourself.

One thing I know the formula is useful for is for improving nth term approximations because the closer you can get your nth term formula in place of the prime function to the resemble the graph above the closer you are to understanding how to at least partially mimic the behaviour of the Primes which in-turn could help you to improve your pi(x) approximations. I know you've followed the crowd and looked at things with far bigger Big O's but 100 years ago you couldn't compute with as bigger number crunches as you can now with computers such that you can now test long equations quickly. Am I crazy.

 

[qpwimblik]

Better yet.
From my studies of this formula you might be able to find Pn's ever further Pn's apart so long as the nth gaps in the sequence are all the same size so you could find Pn sequences with massive gaps. meaning that you might be able to find big Pi's say pi(10^600) very fast.

 

[qpwimblik]

The logic behind the formula

First we have have pattern 1
1,1,2,1,2,3,1,2,3,4,1,2,3,4,5 ... or A002260

we'll call this pattern A(n)

And we have pattern 2
1,2,1,3,2,1,4,3,2,1,5,4,3,2,1 ... or A004736

we'll call this pattern B(n)

Here's how we extrapolate n from A(n) and B(n).

n = ((A(n) + B(n) - 1) (A(n) + B(n))) / 2 - (B(n) - 1)

Now I replace A(n) and B(n) with Prime(n) and Prime(n+1) and see what output I get using the formula above.
Not satisfied with the usefulness of the results (I'm sure anyone who's even tried to think knows that moment)

Anyway yes so I decided to minus the formula from itself with a sequence of values of Pn to see how the distribution would fair with my best results coming from minusing the minused output from a minused output using a sequence of Pn's, what you see in the initial post is the Simplified formula.

sorry it's a bit complicated

 

[qpwimblik]

Red = The main sequence of 8 Primes of Pn being dealt with using the main equation
Blue = The same sequence as Red but out by a small amount between 0 and +7 for each value in the sequence

note. keep a close eye on Reds activities at 0 and the Difference between Red and Blues lines just above zero.

http://www.pyramidus.org/Images/DistributionOfIntrest.jpg

I think we should call my discovery the Wimblik Equlibrium

 

[qpwimblik]

The formula for assessing 3 primes at a time is this.

1/2*(3*prime(n)^2-4*prime(1+n)^2-prime(n)*(3+2*prime(1+n))+(-3+prime(2+n))*prime(2+n)+2*prime(1+n)*(3+prime(2+n)))

And the formula to get very close to the lines the distribution makes is this with e use here being the number e.

lineNumber*0.98*(1-1/2*n*(1+n)-(2+e)*ln(n)+1/2*(-1+n+(2+e)*ln(n))*(n+(2+e)*ln(n)))

which all looks like this.

http://www.pyramidus.org/Images/OnTheLine.jpg

 

[qpwimblik]

Using xlogx instead of Pn

http://www.pyramidus.org/Images/xLogx.jpg

 

[qpwimblik]

A function iterating in on itself.

watch near zero.

http://www.pyramidus.org/Images/BigDistribution.jpg

 

[qpwimblik]

An interesting signature look at the primes distributed.

http://www.pyramidus.org/Images/Complex.jpg

 

[qpwimblik]

An attempt to guess the distribution with a random Integer variable in the formula.

http://www.pyramidus.org/Images/ComplexEq.jpg

 

[qpwimblik]

My VIP find (Rudimentary pattern in the distribution of Primes)

http://www.pyramidus.org/Images/VIPDistribution.jpg

noted to myself must spend a year animating that.

 

[qpwimblik]

Different Random number generation methods seem to exhibit different regularities here's one
example.

http://www.pyramidus.org/Images/NotRandom.jpg

 

[Norwegian]

Since I like the challenge of understanding other peoples thoughts and ideas, I have spent some minutes looking at the posts above.

So we start with a huge expression, and what I assume is its graph, for values of n up to 30000. It is not entirely clear to me what is meant by abs(prime(n)), though I am able to make a guess. Also, the expression can be considerably tidied up by introducing some short notation for repetitive sub-expressions.

I do not get much out of the next couple of posts, then, in the 4th, some explanation is finally revealed. Two sequences are given, and there is a formula for going from its values, and back to n. The two sequences are replaced by prime(n) and Prime(n+1), but the same formula is used to produce some number, and the big expression in the first post is a more involved version of this formula. I still wonder what the significance of this might possibly be.

Then on the next picture, you mention the Wimblik equiulibrium, without clearly stating what that is. This is followed by more graphs. I am sorry, but I think I have failed to understand this thread. On to the next.

 

[qpwimblik]

I was just quickly probing Formulae to look for signature in the nth Prime distribution and I found it. So I know now that the distribution is not random just Infinitely dynamic which does mean that it's !Hard to even approximate which most think is so, anyway Hard does not equal impossible.

 

[qpwimblik]

this is my Christmas Primes distribution analysis.
http://www.pyramidus.org/Images/ChristmasPrimes.gif