I would like to ask this soft question about prime numbers. I believe that (if I'm right) that the more largest known prime number, with few exceptions, (has the form) is a Mersenne prime $$2^p-1,$$ that is a prime number for some large integer $p$.
Thus if one wants to know what is the more largest prime known in science one should be ask to the Great Internet Mersenne Prime Search project, see this Wikipedia. Thus if tomorrow (or the next week, month, year...) if a new largest prime is discovered, surely it is a new Mersenne prime.
Fact. There exist a huge quantity of prime numbers in the gap between two (large, say us the penultimate and the last Mersenne prime known) consecutive Mersenne primes.
Question. I would like to know what is the purpose and/or intention, if such question can be elucidated, to compute those prime numbers in the interval defined by two large and consecutive Mersenne primes. Many thanks.
Is there any objective need to calculate such primes in this interval?
To clarify my question, to know large prime numbers have applications, for example in the real life the RSA codification. And I am asking about what was the intention and/or purpose (scientific, applications,...) to compute the primes numbers, and the quantity of such, that contains a gap between two consecutive Mersenne primes (you can thus evoke that our pair of Mersenne primes are moderately large).