Is $k^{2018}+2018$ prime for some positive integer $k$ ? If yes, which $k$ is the smallest?

Solution 1:

Not sure why this question has no answer since comments contain one, but here are couple more values found by brute force search in Maple. Smallest (probable) primes of these form are given by:

$$ k=129735,145563,147165,\dots $$

Also the leading coefficient is positive, the polynomial is irreducible (by Eisenstein for example) and values it represents are coprime (by $\gcd(f(2),f(3))=1$ for example). Thus it satisfies conditions for Bunyakovsky conjecture, by which it should represent infinitely many primes.

Bonus: Since we are already in $2019$, here are couple examples for $k^{2019}+2019$:

$$k=16294,36688,42188,\dots$$