If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all? Thanks!

If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all?

EDIT: I guess I should add that heuristically, for sufficiently large $m$, the set is so sparse that the expected number of primes of the form $A(m,n)+c$, for fixed $m$ and $c$, should be finite! So maybe I should ask if there is any reason why the number of primes of the form $A(m,n)+c$, for fixed $m$ and $c$, despite $m$ being large, might actually not be finite.

Thanks!