I think there are infinitely many, I work to reach it.

I still believe that the Lenstra-Pomerance-Wagstaff conjecture will be proved, leading to an infinite number of Mersenne primes.

This 1989 article is an interesting introduction: https://www.jstor.org/stable/2323195?seq=1#metadata_info_tab_contents

Thanks Rineke Verbrugge for your valuable comments. This article is valuable.

Dear colleagues,

It is not even known whether the set of Mersenne primes is finite or infinite. For more information on these primes and related references and links, see Sloane’s sequences OEIS A000668 (Mersenne primes - of form 2^p - 1 where p is a prime) and A000043 (Mersenne exponents – primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime), available at website https://oeis.org/ of The On-Line Encyclopedia of Integer Sequences.

,

In my recent preprint available at https://arxiv.org/pdf/1901.07882.pdf

(preprint arXiv, January 2019; a draft version of this paper is available at https://www.researchgate.net/publication/329844912_GOLDBACH-TYPE_CONJECTURES_ARISING_FROM_SOME_ARITHMETIC_PROGRESSIONS),

I have proved an interesting statement (Proposition 1.4) involving the proposed question by Dr. K. Hussein A., the same question about Fermat primes and one property of arithmetic progression 2, 5, 8, 11, 14,...