Wagstaff prime

A Wagstaff prime $p$ is a prime number of the form ${\displaystyle \frac{2^{2n+1}+1}{3}}$. The first few are 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, etc., given in A000979 of Sloane’s OEIS. The exponent of 2 in the formula given above must be an odd number for the result to be an integer in the first place, and that exponent must not be composite.

Currently, the largest known Wagstaff prime, corresponding to an exponent of 42737, is approximately $4.383322622\times {10}^{12864}$. As there is no known special primality test for Wagstaff primes, François Morain had to use elliptic curve primality proving (ECPP) over several months to prove the primality of this Wagstaff prime, finishing in August 2007. (Morain names these primes after Samuel Wagstaff, Jr.) A000978 lists nine exponents which give probable primes.