In the comments to the question: If (an+n)∣(bn+n) for all n, then a=b, there was a claim that 5n+n is never prime (for integer n>0).
It does not look obvious to prove, nor have I found a counterexample.
Is this really true?
Update: 57954+7954 has been found to be prime by a computer: http://www.mersenneforum.org/showpost.php?p=233370&postcount=46
Thanks to Douglas (and lavalamp)!
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 5n+n is 3-probable prime when n=7954. Moreover, for every n less than 7954, we have 5n+n is composite.
To actually certify that 57954+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: 57954+7954 is officially prime. A proof certificate was given by lavalamp at mersenneforum.org.