I still search without success for a prime factor of the huge number $$2\uparrow \uparrow 4+3\uparrow \uparrow 4$$
Another way to write this is $$3^{3^{3^3}}+2^{2^{2^2}}=3^{3^{27}}+65536=3^{7625597484987}+65536$$ Since it can be written as $a^2+3b^2$ with coprime positive integers $a,b$ , we can conclude that a prime factor must have residue $-3$. Since $3$ is obviously impossible , the prime factors must have the form $3k+1$. According to my search there is no prime factor below $3\cdot 10^{11}$ (doublechecks are of course welcome!)
Is there any other restriction using the large exponent $3^{27}$ ? Or can we do no better than to check all primes of the form $3k+1$ until we find a factor ?