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

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)!

Answer

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.

Attribution
Source : Link , Question Author : Aryabhata , Answer Author : Douglas S. Stones

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