I searched prime factors of the numbers
$$z(m,n) := 2 \uparrow \uparrow m + 3 \uparrow \uparrow n$$
where $m,n\ge1$
Interestingly, z(3,3) is prime, the largest prime I found so far and probably the largest of the desired form.
So far, I did not find a prime p, such that $p^2$ divides z(m,n) for some m and n. Since I searched upto about $10^6$, I came to the conjecture that z(m,n) is always squarefree. Has anyone an idea how this can be proven ?