[seqfan] Re: Smallest k>1 such that k^n+n is prime

Joseph Myers jsm at polyomino.org.uk
Mon Apr 16 21:23:32 CEST 2018


On Mon, 16 Apr 2018, Hugo Pfoertner wrote:

> SeqFans,
> 
> usually I now refrain from inventing "is prime" sequences, but a friend
> asked me, for which n there are no solutions for k^n+n=prime. The values of
> n for which I could not find a matching k were n=4, 27, 64, next > 170. If
> we say, k=0 if no such k exist, what are the next expected positions where
> this happens?

If n is of the form 4m^4 there are no such primes by virtue of the 
factorization of a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab) (Sophie 
Germain's identity).

If n is a pth power divisible by p for p odd then the factorization of the 
sum of two pth powers applies.

-- 
Joseph S. Myers
jsm at polyomino.org.uk



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