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[PrimeNumbers] Re: another attempt at Twin prime conjecture...

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  • Jens Kruse Andersen
    ... Definitely no. The only comparable prime quantities are about the _expected_ number of primes based on unproven heuristics. Even if this expectation
    Message 1 of 7 , Nov 6, 2004
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      Suresh Batta wrote:

      > Can we now not say that, since both the ranges have comparable prime
      > quantities, if a P+3 has a twin prime, then squared range
      > ((p+2)^2, p^2) should have too have a twin prime?

      Definitely no. The only "comparable prime quantities" are about the _expected_
      number of primes based on unproven heuristics. Even if this expectation could be
      proven reasonably accurate, it would not say anything about the number of twin
      primes.

      p/log p is the approximate number of primes below p.
      The prime number theorem says it is asymptotically right. This means the
      relative error (NOT the absolute error) tends to 0 when p tends to infinite.

      However, as Decio notes, this and better known approximations are too poor to
      say anything about the number of primes (let alone twin primes) from p to p+x
      when x is much smaller than p.

      It can only be used to say things like:
      The _average_ number of primes in an interval of x numbers _around_ the size of
      p is approximately x/log p.

      Little is known about how large the deviation from such averages can be.
      To illustrate possible deviations, here are the most extreme values known around
      100 digits:

      There are 11 primes (0.16 expected) among the 104-digit numbers p to p+36 for
      p = 24698258*239# + 28606476153371, found by Norman Luhn and I.

      There are 0 primes (30 expected) among the 93-digit numbers c to c+6378 for
      c = 5629854038470321802219554908853741163682800524507382035301697914566243\
      83980052820124370178769, found by Torbjörn Alm and I.

      There probably exists far more extreme cases.

      As far as twin primes go, there is not even anything known about averages.

      --
      Jens Kruse Andersen
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