# Is there a prime of the form $11^k+k^{11}\$?

Is there a natural number $k\ge 1$, such that $11^k+k^{11}$ is prime ?

I checked the numbers upto $k=3000$ and did not find a prime number. On the other hand, for $k=76$ and for $k=142$, there is no small prime factor (http://factordb.com/index.php?query=11%5En%2Bn%5E11&use=n&n=1&sent=Show&VP=on&VC=on&EV=on&FF=on&perpage=200&format=1)

I cannot find a reason that $11^k+k^{11}$ must be composite for every $k\ge 2$ , for example because we can show that algebraic or aurifeuillian factors must exist.

You did not check quite far enough. $11^{3100}+3100^{11}$
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• That is true. The number has $3,229$ digits , so a rigorous proof will take a while. But the number passes many SPRP-tests, so it is without doubt prime. It is curious, that the number of digits is prime as well. – Peter Jun 22 '16 at 15:14