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Re: [PrimeNumbers] Need help-what is Chebyshev's theorem about the distributi...

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  • mikeoakes2@aol.com
    In a message dated 04/06/04 05:54:15 GMT Daylight Time, ... I think you must be thinking of Bertrand s postulate, which is that, for every n 3, there is a
    Message 1 of 1 , Jun 4, 2004
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      In a message dated 04/06/04 05:54:15 GMT Daylight Time,
      billroscarson@... writes:


      > It has been suggested to me that Chebyshev (I am not sure which one-
      > apparently there are many)has a theory that says something like
      > there is always one prime between n^2 and (n^2+2n). I cannot find
      > it, or I don't recognize it when I do find it. Can anyone tell me if
      > the theorem is as I stated, and where I can find it documented?
      > Thanks for any help.
      >

      I think you must be thinking of Bertrand's postulate, which
      "is that, for every n > 3, there is a prime p satisfying n < p < 2n-2.
      Bertrand verified this for n < 3,000,000 and Tchebychef proved it for all n > 3 in
      1850."
      [Hardy & Wright (1979), p. 373]

      If p[n] is the nth prime number, define
      d[n] = p[n+1] - p[n]
      Then your statement would be equivalent to the assertion that d[n] <=
      2*p[n]^0.5.

      This is probably true (for n "sufficiently large"), but has never been proved.
      In fact, it is widely conjectured that d[n] < p[n]^0.5 except for a finite
      number of cases.

      If the Riemann hypothesis is assumed, it is (relatively!) easy to prove that,
      if t is any fixed number > 0.5, then d[n] < p[n]^t except for a finite number
      of cases.

      By 1990 the strongest result unconditionally proved (by Mozzochi in 1986) was
      that this statement is true with t any fixed number > 1051/1921.
      [A..E.Ingham "The Distribution of Prime Numbers" (1990)]

      Is this the latest state of play, anyone know?

      -Mike Oakes



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