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

[Allan Wechsler]

K^4 + 4 factors algebraically into (K^2 - 2K + 2) (K^2 + 2K + 2).

K^27 + 27 factors into (K^9 + 3) (K^18 - 3K^9 + 9).

Similarly, K^64 + 64 = (K^32 - 4K^16 + 8) (K^32 + 4K^16 + 8).

See https://oeis.org/A097792 for more information; the next K with no prime
K^n + n is 216.



On Mon, Apr 16, 2018 at 2:23 PM, Hugo Pfoertner <yae9911 at gmail.com> wrote:

> n=18^2=324 seems to be a likely candidate for the next occurrence of no
> solution, checking k<10^6.
>
> On Mon, Apr 16, 2018 at 8:13 PM, Harvey P. Dale <hpd at hpdale.org> wrote:
>
> > Hugo:
> >         I now see that k must be greater than 1 (from the subject line,
> > rather than the text), so please ignore my post.
> >         Best,
> >         Harvey
> >
> >
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Harvey
> > P. Dale
> > Sent: Monday, April 16, 2018 2:13 PM
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Subject: [seqfan] Re: Smallest k>1 such that k^n+n is prime
> >
> > Hugo:
> >         Doesn't k=1 work for n=4?
> >         Best,
> >         Harvey
> >
> >
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Hugo
> > Pfoertner
> > Sent: Monday, April 16, 2018 1:33 PM
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Subject: [seqfan] Smallest k>1 such that k^n+n is prime
> >
> > 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?
> >
> > A corresponding sequence from n=1 through n=65 would be:
> > 2, 3, 2, 0, 2, 175, 16, 3, 2, 539, 32, 221, 118, 417, 2, 85, 14, 133, 22,
> > 81, 76, 115, 12, 55, 28, 15, 0, 2465, 110, 31, 232, 117, 230, 3, 12, 851,
> > 4, 375, 2, 1599, 48, 5461, 46, 15, 218, 6815, 78, 7, 100, 993, 28, 901,
> > 624, 13, 252, 183, 226, 43247, 104, 5063, 1348, 777, 1294, 0, 1806,
> >
> > Hugo Pfoertner
> >
> > --
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> >
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> >
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