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October 12th, 2016, 08:26 AM   #1
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Hi everyone! I am new to the forum.

In my spare time, I've been trying to find a way to count twin primes under any given x. I know there is a method that exists that utilizes the Twin Prime Constant. Anyways, I found the distribution of twin primes to be related to the Euler-Mascheroni constant and it took me by surprise.

To estimate how many twin primes there are between consecutive squares of primes [P(n+1)^2 - P(n)^2] I did the following:

[Prime(n+1)^2 - Prime(n)^2] / [(Product of all primes up to n)/(Product of all Prime-2 up to n, starting with P=5)]

Take that result and multiply it by (1/4) * e^(2*(Euler-Mascheroni Constant)) = 0.79305474

So for example, to estimate the amount of twin primes between 13^2 and 17^2

= [17^2 - 13^2] / [(2*3*5*7*11*13)/(3*5*9*11)]

= 120 / 20.222

= 5.934

= Take that result and multiply it by 0.7905474

= 4.71 Twin Primes between 13^2 and 17^2

I tried to insert a table to show you how close my estimate is to the actual number of twin primes, but could not figure it out. Anyways, here is a link to the table:

Math: Twin Primes - Album on Imgur

What are your thoughts on this? Is it directly related to the Twin Prime counting function that already exists?

Last edited by skipjack; October 12th, 2016 at 10:38 AM.
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