--- In primeform@yahoogroups.com, "andrew_j_walker" <ajw01@...> wrote:
>
> --- In primeform@yahoogroups.com, "masserto" <masserto@y...> wrote:
> >
> > Dear Guido,
> >
> > I recently found that 2*k*3^n-1 is always composite if
> >
> > k=739171331147778631
> >
> > The "covering set" for this Riesel number is:
> >
> > {5,7,13,17,19,37,73,97,577,757,769}
> >
> > Has anyone found a smaller k so that 2*k*3^n-1 is always composite?
> >
> > Best regards,
> > Tom Masser
> >
>
> Nice work, that's smaller than any I remember seeing. Have you looked
> at the plus case?
>
> Andrew

There has been a delay since this was posted.

Regarding the Sierpinski case:

One such covering set is [13,5,7,41,73,17,193,6481,97,577] which have
multiplicative order base 3 of 3,4,6,8,12,16,16,24,48,48, all of which
are factors of 48. Using CRM provides the following k which provides
composite 2*k*3^n+1 for all n:

36785490291994693

I am by no means convinced this is the smallest k but it might be as
it is 20 times smaller than the lowest known Riesel. It will be a very
hard problem to prove this is the lowest 2*k never prime.

The corresponding Riesel associated with this covering set provides a
larger value of k than the Tom's Riesel value.

A nice series for OEIS would be 78557,36785490291994693,66741,159986....

Mooted Sierpinski numbers base a=2,3,4,5..., where to be the k value
the Sierpinski must be multiplied by all primes which have
multiplicative order base a of 1 (to elimiate trivial results). Anyone
up to extend this series as a challenge?

Regards

Robert Smith