double Mersenne number

A double Mersenne number is a number of the form $2^{2^p-1}-1$. Put another way, negative one plus 2 raised to the power of a Mersenne number. The first few double Mersenne numbers (and the ones small enough to show here) are: 3, 7, 127, 170141183460469231731687303715884105727.

If a double Mersenne number is itself prime, then it is called a double Mersenne prime. Obviously its index is then a Mersenne prime. The four double Mersenne numbers listed above are all primes, but as of today, $2^{170141183460469231731687303715884105727}-1$ is a probable prime, despite an intense effort to find factors. (According to the Prime Pages, if it’s composite, its least prime factor must be at least $5\times {10}^{51}$.