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25173About Sierpinski's Theorem

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  • Jose Ramón Brox
    Jun 6 5:20 PM
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      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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