[seqfan] Re: Smallest k > 1 such that n^k+k^n is prime, or 0 if no such k exists

israel at math.ubc.ca israel at math.ubc.ca
Fri Dec 11 22:17:12 CET 2015


When k == 1 (mod 4), 
4^k + k^4 = 4*x^4 + k^4 = (2 x^2 - 2 k x + k^2)(2 x^2 + 2 k x + k^2)
where x = 4^((k-1)/4).  
When k == 3 (mod 4),
4^k + k^4 = 64*x^4 + k^4 = (8 x^2 - 4 k x + k^2)(8 x^2 + 4 k x + k^2)
where x = 4^((k-3)/4).

Cheers,
Robert

On Dec 11 2015, Felix Fröhlich wrote:

>Sequence (with offset 2) starts 3, 2, ?, 24, ?, 54, 69, 2 .....
>
>Question: Is a(4) = 0?
>
>What I found is:
>- when k is even, 4^k + k^4 is also even
>- when k is odd and not a multiple of 5, the last digit of 4^k is 4 and the
>last digit of k^4 is 1, so  4^k + k^4 is a multiple of 5
>
>This leaves the cases where k % 10 = 5. Can such a value for k exist such
>that 4^k + k^4 is prime?
>
>Similar question for n = 6: Can an eligible k exist? Or are there proofs
>that a(4) = 0 and a(6) = 0? If those questions can be answered, then maybe
>this could be added to the OEIS.
>
>Best regards
>Felix
>
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