Hi

In a separate thread, I stated that when a prime p is the sum of two natural numbers, each in the range (2, (p+1)/2)

say p=a+b , a and b belong to this range,

then a and b must be relatively prime.

When p is not prime, there are random cases where (a,b) are relatively prime

to each other, or not.

The range must be such to prevent counting twice.

I think there is an Euler function that does the counting of relative primes within a certain range.

There is no exact formula to count primes exactly, only approximate for large n

It might be useful to actually understand what these formulas might mean and why there isn't a formula for the twin primes, yet, for instance. For the primes such a formula is known since Riemann and is known, indeed, as the ``explicit formula''-though it's, actually, less explicit than the name suggests, since it contains the zeros of the Riemann zeta-function; that's why it's useful to know them, at all!

These lecture notes: http://people.maths.ox.ac.uk/greenbj/papers/primenumbers.pdf might be a good starting point for understanding the explicit formula.

On the twin prime conjecture, this might be useful: https://arxiv.org/pdf/1910.14674.pdf

Yes it will be very helpfull thanks. I was using fornula X/ ( 3 * 2^log X). I Tested it until 50000 and it was great. I checked with big numbers and it is not usefull. İt has so big difference from twin primes count. Thank you

Dear Muraqt Kılıç

I hope you find the following article is helpful.

It includes an elementary algorithm to count prime numbers.

http://m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf

Regards