If M(p) = 2^p-1, then M(M(p)) is called

**double Mersenne number**, and if this number is prime, then it is called

**double Mersenne prime**, M(M(p)) is prime for p = 2, 3, 5 and 7, but not for all 11<=p<=59, and the status is unknown for p=61. Now, we consider the Wagstaff number W(p) = (2^p+1)/3 for odd prime p, then W(W(p)) is called

**double Wagstaff number **, and if this number is prime, then it is called

**double Wagstaff prime**, it is known that W(W(p)) is prime for p = 3, 5 and 7, but not for all 11<=p<=29 (the p=23 case is divisible by 129469791307, see

factordb), but how about p=31 or above? Are there any double Wagstaff primes > W(W(7))? (related to the conjecture that there are no double Mersenne primes > M(M(7)))