What is the smallest prime factor of the number $14^{14^{14}}+13\ $?

Question

What is the smallest prime factor of the number

$$N\ :=\ 14^{14^{14}}+13\ ?$$

The number of digits of $N$ is $12,735,782,555,419,983$ (The number of digits of $N$ has itself $17$ digits). The first digits of $N$ are $1698324865652...$ and the last digits are $...6015154651149$

It is hopeless to apply a primilaty test for $N$ because $N$ is far too large. I applied trial division and found no prime factor below $6\times 10^9$.

I do not think that there are any better methods to find a factor of such a number than trial division, so I invite number-theory-enthusiasts to join in the search for prime factors.

Answer 1

The smallest prime factor of $ 14^{14^{14}}+13$ is $13990085729$.

Trial division should be the only way to find factors of such big numbers.

Pollards-RHO Method and ECM both need $gcd(q,N)$ for some small $q$ while $N$ is far out of range for such computations.

The trial division with modular exponentiation I used in Pari/GP:

forprime(p=2,10^11,if(!(Mod(14,p)^11112006825558016+13),return(p)))

returns the result

time = 10min, 58,230 ms.
%1 = 13990085729

Note: precomputing $14^{14}=11112006825558016$ speeds up the algorythm about 7%.