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About Sierpinski's Theorem

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  • Jose Ramón Brox
    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
    Message 1 of 3 , Jun 6, 2013
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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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    • djbroadhurst
      ... What if we ask for 3^n·k+1 or more generally, for a base b^n·k+1? http://www.mersenneforum.org/showthread.php?t=9738 David
      Message 2 of 3 , Jun 7, 2013
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        --- In primenumbers@yahoogroups.com, Jose Ramón Brox <ambroxius@...> wrote:

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

        http://www.mersenneforum.org/showthread.php?t=9738

        David
      • Jose Ramón Brox
        Thank you both for the info! Jose 2013/6/7 djbroadhurst ... -- La verdad (blog de raciocinio político e información
        Message 3 of 3 , Jun 7, 2013
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          Thank you both for the info!

          Jose


          2013/6/7 djbroadhurst <d.broadhurst@...>

          > **
          >
          >
          >
          >
          > --- In primenumbers@yahoogroups.com, Jose Ram�n Brox <ambroxius@...>
          > wrote:
          >
          > What if we ask for 3^n�k+1 or more generally, for a "base" b^n�k+1?
          >
          > http://www.mersenneforum.org/showthread.php?t=9738
          >
          > David
          >
          >
          >



          --
          La verdad (blog de raciocinio pol�tico e informaci�n
          social)<http://josebrox.blogspot.com/>


          [Non-text portions of this message have been removed]
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