Analogues to Fermat primes related to Pell's equation

The coefficients of some weight 3 modular forms give reason to study primes of the form p = 2x(2) - 1 = 2dy(2) + 1. If x(a), y(a) are the positive solutions of Pell's equation x(2) - dy(2) = 1, given by x(a) + y(a)root d = (x(1) + y(1)root d)(a), and if p(a) = 2x(a)(2) - 1 is prime, then a = 2(m) is a power of 2. So there are analogues to the Fermat numbers 2(a) + 1.