This is a  question  about generalizations of the twin primes conjecture.

I would  like to know a counterexample, or a  proof,  for the  following cpuple or related  arithmetical sentences:

$(\forall p)(\exists n)(\exists q)( Prime(p) \wedge  Prime (q) \wedge (p = 2^n + q))$

In informal words, every prime number $p$  can be  written  as the sum of a smaller  prime $q$ and  a    power  of  two.

$(\forall q)(\exists n)(\exists p)  ( Prime(p) \wedge  Prime (q) \wedge (p = 2^n + q))$

In informal words, for  every prime number $q$  there exists a number  $n$  such that the sum  of  $q$  and the  $n$-th power  of  two is a  prime number.