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Re: A New Conjecture?

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  • djbroadhurst
    ... Noting the counterexamples for N = 125 and N = 126 that Jack has given, I remark on the more general problem of proving that, for some fixed theta, there
    Message 1 of 7 , Dec 28, 2009
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      --- In primenumbers@yahoogroups.com,
      "ronaldpeterdwyer" <dwyer_ron@...> conjectured:

      > For any positive integer N>3, there is a prime number
      > between N and N-(sqrt N).

      Noting the counterexamples for N = 125 and N = 126
      that Jack has given, I remark on the more general
      problem of proving that, for some fixed theta,
      there is always a prime between (N - N^theta) and N,
      for sufficiently large N (depending on theta).

      There is a proof for theta = 21/40 :

      R. C. Baker, G. Harman, and J. Pintz,
      The difference between consecutive primes, II,
      Proc. London Math. Soc., 83 (2001), 532-562.

      If one assumes the Riemann hypothesis,
      then a similar result follows for theta > 1/2.
      I know of no such argument for theta = 1/2.

      It has, of course, long been *conjectured*
      that such a result obtains for any theta > 0.
      [Ribenboim attributes this conjecture to Piltz, in 1884.]

      David
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