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

Question

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.

I experience a deja-vu... There must be a glitch in the Matrix ! — Lucian, May 31 '15 at 12:35
As most commoners would say, "Since there are no primes to $n=1500$, then $2n^n+91$ must therefore always generate composites for all integers $n \geq 1$" — Trogdor, May 31 '15 at 12:44
There are two cases: $n=3k$, and $n=6k-1$. Otherwise, $2n^n+91$ cannot be prime, since all primes greater than $3$ are of the form $6m\pm1$. — Lucian, May 31 '15 at 12:47

score 2 · Accepted Answer · answered May 31 '15 at 12:37

2

$$2 \times 1949^{1949} + 91$$

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 $n$.

answered May 31 '15 at 12:37

TonyK

64,559

How did you conclude that it is 'probably prime'? But it does look suspiciously prime... – Trogdor May 31 '15 at 12:45
@Trogdor there are plenty of test that show a number is composite, if they fail it is "probably prime." See http://en.wikipedia.org/wiki/Probable_prime – quid May 31 '15 at 12:59
I like "suspiciously prime". All we have to do is decide on a meaning for it. – TonyK May 31 '15 at 13:05
You're the one who made me *recant* my skeptical statements about proving a large number prime. Same person asking, too. They deleted my revised answer, which is a shame. http://math.stackexchange.com/questions/402357/is-184720132-really-a-prime – Will Jagy May 31 '15 at 15:32
@WillJagy: Philistines. – TonyK May 31 '15 at 16:18

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

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