Checking the prime factors of the numbers
$$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$
a,b positive integers with a < b
the number
$$z(14,15) = 14 \uparrow \uparrow 3 \ +\ 15 \uparrow \uparrow 3 = 14^{14^{14}}+15^{15^{15}}$$
turned out to be a candidate surviving trial division up to $10^{10}$.
Of course, $z(14,15)$ still has a very low chance to be prime, so I search a prime factor. Any ideas how the search can be accelerated ? Can ECM or similar methods be used to find factors of numbers of this magnitude (about $10^{10^{18}}$) ?