primes of the form $ 2^p +q $ where p and q are primes i am looking some refernce or texttes about prime numbers of the form
$$ p =2^{q}+r $$ 
where all $p,q,r $ are prime numbers
there are several examples $ 11= 2^3+3 $ , or $ 2^{4}+7 =23 $

If $r$ is a dual Sierpinski number, for example $r = 271129$ which is also prime, then there are no solutions to $p = 2^q + r$ with prime $p$.  There is no value of $r$ that is known to have an infinite number of solutions.
The k-tuple conjecture implies that there are an infinite number of prime solutions $(p, r)$ for any fixed $q \gt 0$.  But it seems like Zhang's theorem and related results do not help us show that there exists even one such $q$.
Is it possible to prove there are an infinite number of satisfying triples?
