Thanks Peter.

I did not say I have found a proof to Mersenne number. I want to know more about prime number. I am also interested in prime gaps. Please, lead me to those strong recent theorems on prime gap.

Thanks

Okay. There is no proof yet. The Mersenne formula can only get a fraction of the infinitely many prime that exist.

https://en.m.wikipedia.org/wiki/Formula_for_primes

You can try 3^p - 2 to generate some prime numbers. It looks like Mersenne numbers but it is faster.

This formula generates 7, 79, 241,727, 6559, ...

I will come back on this subject.

You  are right. This is why I wrote I will come back.

The general formula are :  q^p-(q-1)  and  q^p-(q-2)  till  q^p -1

It opens a new field in the study of prime numbers and allows generalization of some theorems.

Thanks Profs Mohamed and Peter.

Your contributions were useful.