5n+n5^n+n is never prime?

A general rule-of-thumb for “is there a prime of the form f(n)?” questions is, unless there exists a set of small divisors D, called a covering set, that divide every number of the form f(n), then there will eventually be a prime. See, e.g. Sierpinski numbers.

Running WinPFGW (it should be available from the primeform yahoo group http://tech.groups.yahoo.com/group/primeform/), it found that $5^n+n$ is 3-probable prime when n=7954. Moreover, for every n less than 7954, we have $5^n+n$ is composite.

To actually certify that $5^{7954}+7954$ is a prime, you could use Primo (available from http://www.ellipsa.eu/public/misc/downloads.html). I’ve begun running it (so it’s passed a few more pseudo-primality tests), but I doubt I will continue until it’s completed — it could take a long time (e.g. a few months).

EDIT: $5^{7954}+7954$ is officially prime. A proof certificate was given by lavalamp at mersenneforum.org.