Let $p_n$ be the $nth$ prime and define $p_n^{(m)}$ by $p_n^{(1)}=p_n$ and $p_n^{(m+1)}=p_{p_n^{(m)}}$:
$p_n^{(2)}=p_{p_n}$, $\;p_n^{(3)}=p_{p_{p_n}}$ and so far...
For some coprime numbers $a,b$, not both odd, do there exist a number $m\in \mathbb Z_+$ such that $A(a,b,m)=\{ap_n^{(m)}+b\in\mathbb P|n\in\mathbb Z_+\}$ is finite? $\mathbb P$ is the set of primes.
I have done some tests and there is no sign of finiteness.
Conjecture:
If $a$ and $b$ are coprime and $a+b$ is odd, then $A(a,b,m)$ is infinite for all $m\in\mathbb Z_+$.
One consequence of the conjecture, which I don't expect to be proved, is that any even gap just not only exists but exists infinitely, not only for primes, but also for the case when one of the primes is of the form $p_n^{(m)}$ any $m$. So perhaps there is a possibility of counterexamples?
See also this paper.