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Re: perfect numbers and twin primes

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  • jbrennen
    ... I am with David on this one. The heuristics argue against any other such examples. Essentially, one needs to have the following constellation of four
    Message 1 of 4 , Oct 23, 2002
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      --- "David Broadhurst" <d.broadhurst@o...> wrote:
      > > infinitely many perfect numbers p, such
      > > that both p+1 and p+3 are prime
      >
      > i bet on just a single pair {29,31}

      I am with David on this one.

      The heuristics argue against any other such examples.

      Essentially, one needs to have the following "constellation"
      of four primes:

      p
      2^p-1
      (2^p-1)*2^(p-1)+1
      (2^p-1)*2^(p-1)+3

      Very roughly speaking, the chance of each of these being prime is:

      1/log(p)
      1/(p*log(2))
      1/(2*p*log(2))
      1/(2*p*log(2))

      Multiply them together:

      1/(4*log(2)^3*p^3*log(p))

      After exhaustively testing with small values of p (p<100), the
      resulting sum of expected prime 4-tuples (for p>100) is rather
      discouraging, even when one "corrects" the probabilities above
      appropriately.
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