Hi.

Given the following sequence (of Mersenne primes):

$ A_{1} = 2 $

$ A_{n} = 2^{A_{n-1}} - 1 $

The first five elements are all prime numbers:

$ 2 $

$ 2^{2}-1=3 $

$ 2^{3}-1=7 $

$ 2^{7}-1=127 $

$ 2^{127}-1=170141183460469231731687303715884105727 $

As far as my memory serves, it has been conjectured that this sequence contains ONLY prime numbers.

Sadly, calculating the sixth element (let alone proving it's a prime number or providing a counterexample) appears computationally infeasible, so I guess that this remains an open conjecture.

My question is with regards to a similar type of sequences:

$ B_{n} = 2^{B_{n-1}} - 1 $

Where $ B_{1} $ is a Mersenne prime which is NOT in $ A_{n} $

Has it been conjectured that ANY such sequence will ALWAYS contain a composite number?

Thanks