Hi all!

I recently got interested on Sierpinki's Theorem: Consider the sequence

S_k={2^n·k+1} where k is a fixed positive integer. Then there exist

infinitely many k's such that S_k contains no primes.

I wonder about generalizations and related results:

a) What if we ask for 3^n·k+1 or more generally, for a "base" b^n·k+1?

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?

I hope that some of you have a (maybe partial) answer to any of these

questions!

Kindest regards,

Jose Brox

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