Sierpinski in other bases and infinity
- --- In firstname.lastname@example.org, Jose Ramón Brox <ambroxius@...> wrote:
>b. There are many k which are Sierpinski or Riesel for multiple bases, k=2 being common
> b) Can a number be Sierpinski for several different bases b simultaneously?
> c) Can a number be Sierpinski and Riesel at the same time? (Recall that
> Riesel numbers are as Sierpinski but with the general formula 2^n·k-1).
> d) What can be said about S_k when k is NOT Sierpinski? Does it have
> infinitely many primes, or can it have a finite nonzero number of them?
> [Non-text portions of this message have been removed]
c. Yes: these are Brier numbers..for base 2 see
They exist for other bases
d. The largest number of primes found for any k in the power series to date is 180.
The smallness of this number does not preclude there being a k with an infinite number of primes, and I think such k might exist, given that it is possible (I think) to construct a k such that number of primes in the series k*2^n+/-1 is equal to n. The largest found to date is (I think) n=17. (Wroblewski k= 2759832934171386593519)
I don't think that all k provide either zero (Sierpinski and special case) or infinite primes. There are an infinite number of Sierpinski k, and there an infinite number of k that are non-Sierpinski.
There are a special class of k that have zero primes because of factorisation rules and partial factorisation rules, and are not called Sierpinski numbers, but can be shown to have no primes.
In my opinion, the only way that there cannot be an infinite number of k with a finite, nonzero number of primes is if all non-Sierpinski k fall into the category of having infinite primes. This seems very unlikely to me.
Comments welcome on the above. I'm no expert on infinity.