The partial prime factorization of $$3^{3^3}+4^{4^4}=3^{27}+4^{256}$$ is $$43\times 691\times C150$$ , where
C150 = 451243830308033423066470063548138731446820106370692739801553577347348434357807928408503829911718891653149525735754923799473425194761436743056617460491
is a composite $150$-digit number without a small prime factor.
It is very likely that it has no prime factor with less than $30$ digits because the number passes at least $1000$ ECM-$250$K-curves and at least $1200$ ECM-$1$M-curves.
Can I take advantage of the special form of the number and accelerate the search of further prime factors ?
Does anyone know further prime factors of $3^{27}+4^{256}$ ?
C150 is the product of the two primes, one with 55 (decimal) digits and one with 96 digits,
$P55=1449299471738053389661827008867152641816024786660724327$
and
$P96=3113530633988882752054263646036326764738143136281825948356832545797877\ 39684923341337929165875133$.
I found these factors using the GMP-ECM algorithm incorporated in Sage, with the initial bound B1 set equal to $11\times10^6$. It took the program just over 5 days on a Linux laptop.