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question on (un)boundedness

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  • James Merickel
    Dear group members: Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime
    Message 1 of 3 , Sep 24, 2013
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      Dear group members:
      Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime between b^N(b)-b and b^N(b)?  I imagine I may be able to address this heuristically myself without much bother (and the answer is there is no bound, but of course until I actually try I won't be too sure), but I wonder if there is anything stronger than heuristics on this or it's 'the usual for such number theory questions'.
      Jim
    • Kermit Rose
      ... It is not an easy question to answer. (2^2-2,2^2) = (2,4) (2^3-2,2^3) = (6,8) (2^4-2,2^4) = (14,16) N(2) = 4 (3^2-3,3^2) = (6,9) (3^3-3,3^3) = (24,27)
      Message 2 of 3 , Sep 25, 2013
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        On 9/25/2013 7:03 AM, primenumbers@yahoogroups.com wrote:
        > question on (un)boundedness
        > Posted by: "James Merickel"moralforce120@... moralforce120
        > Date: Tue Sep 24, 2013 11:17 am ((PDT))
        >
        > Dear group members:
        > Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime between b^N(b)-b and b^N(b)? I imagine I may be able to address this heuristically myself without much bother (and the answer is there is no bound, but of course until I actually try I won't be too sure), but I wonder if there is anything stronger than heuristics on this or it's 'the usual for such number theory questions'.
        > Jim

        It is not an easy question to answer.


        (2^2-2,2^2) = (2,4)
        (2^3-2,2^3) = (6,8)
        (2^4-2,2^4) = (14,16) N(2) = 4

        (3^2-3,3^2) = (6,9)
        (3^3-3,3^3) = (24,27) N(3) = 3

        (4^2-4,4^2) = (12,16)
        (4^3-4,4^3) = (60,64)
        (4^4-4,4^4) = (252,256) N(4) = 4
      • James Merickel
        Empirically it appears there is for any n a finite number of values b (both b and n in {2,3,4,...}) such that the prime preceding b^n is not greater than
        Message 3 of 3 , Sep 26, 2013
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          Empirically it appears there is for any n a finite number of values b (both b and n in {2,3,4,...}) such that the prime preceding b^n is not greater than b^n-b, with for reasonably small n an easy-to-find value that makes a good conjecture for the count.  For n=2 it seems certain there are no such, for example, just empirically.  If this finitude is provable it would settle the question asked.
          JGM  

          From: Kermit Rose <kermit@...>
          To: primenumbers@yahoogroups.com
          Sent: Wednesday, September 25, 2013 2:30 PM
          Subject: [PrimeNumbers] Re: question on (un)boundedness
           
          On 9/25/2013 7:03 AM, primenumbers@yahoogroups.com wrote:
          > question on (un)boundedness
          > Posted by: "James Merickel"moralforce120@... moralforce120
          > Date: Tue Sep 24, 2013 11:17 am ((PDT))
          >
          > Dear group members:
          > Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime between b^N(b)-b and b^N(b)? I imagine I may be able to address this heuristically myself without much bother (and the answer is there is no bound, but of course until I actually try I won't be too sure), but I wonder if there is anything stronger than heuristics on this or it's 'the usual for such number theory questions'.
          > Jim

          It is not an easy question to answer.

          (2^2-2,2^2) = (2,4)
          (2^3-2,2^3) = (6,8)
          (2^4-2,2^4) = (14,16) N(2) = 4

          (3^2-3,3^2) = (6,9)
          (3^3-3,3^3) = (24,27) N(3) = 3

          (4^2-4,4^2) = (12,16)
          (4^3-4,4^3) = (60,64)
          (4^4-4,4^4) = (252,256) N(4) = 4

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