Quote:
Originally Posted by sweety439
it is known that W(W(p)) is prime for p = 3, 5 and 7

There are no double Wagstaff primes for p ≤ 23. All of these have known factors. For higher p I know of no PRP tests or even the most basic factoring attempts.
Code:
3 3 prime
5 11 prime
7 43 prime
11 683 1676083,26955961001
13 2731 67399191280564009798331,2252735939855296339250682011
17 43691 349529
19 174763 173085275201
23 2796203 129469791307,36992613766212121
29 178956971
31 715827883
37 45812984491
41 733007751851
43 2932031007403
47 46912496118443
53 3002399751580331
59 192153584101141163
61 768614336404564651
There are no unknown Wagstaff primes below 10M. Ryan Propper in 2013 searched at least some of the space between 10M and 14M but he does not recall exactly which exponents. I have a hunch that there are no undiscovered Wagstaff primes below 14M.