[seqfan] Re: Iterating "smallest odd prime divisor of n^2 + 1"

[Neil Sloane]

> for n=4949, 6051, 9751 and 12281, at least, the map "n -> smallest odd
prime divisor of n^2+1" diverges.

Very nice!  Could you explain the proof?

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Sat, Nov 4, 2017 at 7:08 AM, Luca Petrone via SeqFan <
seqfan at list.seqfan.eu> wrote:

> Dear All,
>
> for n=4949, 6051, 9751 and 12281, at least, the map "n -> smallest odd
> prime divisor of n^2+1" diverges.
>
> Regards,
>
> Luca Petrone
>
> >
> >     Il 4 novembre 2017 alle 11.45 Frank Adams-Watters via SeqFan <
> seqfan at list.seqfan.eu> ha scritto:
> >
> >     It was definitely my impression that it did always fall into that
> loop when I looked at the question in conjunction with my edit to the
> related sequence A031439 back in 06. I didn't have any idea of how to prove
> it, however.
> >
> >     Probabilistically, it seems likely: you only have to get a term
> congruent to 2 or 3 mod 5, and you immediately fall into the loop.
> >
> >     Franklin T. Adams-Watters
> >
> >     -----Original Message-----
> >     From: Neil Sloane <njasloane at gmail.com>
> >     To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >     Sent: Fri, Nov 3, 2017 11:59 pm
> >     Subject: [seqfan] Iterating "smallest odd prime divisor of n^2 + 1"
> >
> >     Dear Seq Fans, While I was at Hofstra Univ. the other day, Zoran
> Sunik
> >     asked, if you iterate the map "n -> smallest odd prime divisor of
> n^2+1",
> >     do you always end in the 2-cycle (5 <-> 13) ?Does anyone know?
> >     See A125256 for the map, and also its bisections A256970and A293958.
> >     If this is true, then there could be a sequence giving the number of
> >     iterations needed to reach the loop, or to reach any loop if there
> are
> >     others ...
> >
> >     --Seqfan Mailing list - http://list.seqfan.eu/
> >
> >     --
> >     Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>