3,5,11 - sequence /proposal for an additional definition

[Gottfried Helms]

Arggh ...

Am 08.11.2005 22:13 schrieb Gottfried Helms:
> Seqfans -
> 
>  1) a few stupid typing errors, sorry. But it doesn't
>     spoil the meaning too terribly, so I'll just leave it as it was.
> 
>  2) The matter can be simplified:
>     the inf-hyperroot n^^°° = a (mod p) can more simply be
>     expressed as
> 
>      n^a = a  (mod p)
> 
>  and so we ask, whether in the residue-group of p an "n"
>  exists with that property, or in other words:
> 
>    whether there exists an "n" which can be expressed as
>    the a'th root of a.
> 
>  The number of solutions n for a given p seems to increase
>  with p, and is even higher if p is not prime but composite.
> 
>  There is always a "trivial" solution, for n=1 (mod p) (obvious).
> 
>  The sequence 3,5,11 is then
>   the sequence of primes>2, which have a non-trivial element "n"
>   where n^a = a (mod p, a integer >0)

**************************** which have *no* nontrivial element "n"
    where n^a =a (mod p, 0<a<p , a integer)

Note also the amazing other relation of 3,5,11, which expresses
the generalized wieferich-property of 11 to base 3:

   3^5 - 1
  -------- = 11^2     // 11 is a co-factor-free generalized wieferich prime base 3
    3 - 1


Gottfried Helms