Does there exist a prime that is a sum of two prime power towers?

Question

Does there exist prime number of the form $$\huge 2^{3^{5^{\,.^{.^{.\,^{p_n}}}}}} + p_n^{p_{n-1}^{\,.^{.^{.\,^{3^{2}}}}}}$$ where $p_n$ is the $n$-th prime number(and both towers are running through the first $n$ primes in order), other than the trivial one $$2^3+3^2=17$$ This number that is the sum of two power towers of first $n$ primes is more than just extremely difficult, it is insanely difficult to become a prime, because if the lower towers divisible by say $k$, then the main number is also divisible by $k$. For example $$\Large 2^{3^5}+11^{7^5} \qquad\qquad 2^{3^{5^7}}+11^{7^{5^3}}$$ $2^{3^{5^{7}}}+11^{7^5}$, are all divisible by $23$, and so is $$\Large 2^{3^{5^{7^{11}}}}+11^{7^{5^{3^2}}}$$ So does there exist a prime number of this form other than the trivial one $2^3+3^2$?

Answer 1

It is quite unlikely there are any more. $2^{3^5}+5^{3^2}$ has many factors, including $7$. The first tower dominates the second in size. Using the chance that a number $n$ is prime is $\frac 1{\log n}$, the first candidate has a chance $\frac{1}{3^{5^7}\log 2} \approx 10^{-37274.93885043456}$ The later terms will decrease so fast the sum will not differ from this by an amount you can see.