TITLE: What is the smallest prime of the form $2n^n+91$?
QUESTION [5 upvotes]: I wondered what the smallest prime of the form $2n^n+k$ is for some odd $k$. For
$k<91$, there are small primes, but for $k=91$ , the smallest prime (if it exists)
must be very large.

What is the smallest prime of the form $2n^n+91$ ?

It is clear that $\gcd(91,n)=1$ must hold. $2n^n+91$ is composite for every
natural number $n$ below $1000$.
$$2\times 15^{15}+91 = 42846499\times 20440124659$$
shows that the smallest prime factor can be large.

REPLY [2 votes]: $$2 \times 1949^{1949} + 91$$
is probably prime! (Running a rigorous primality test on it would take almost a whole day $-$ see here, for instance $-$ so I'm not going to do that.) $2n^n+91$ is composite for all lower values of $n$.