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Re: [PrimeNumbers] Twins

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  • Jack Brennen
    ... Yes, it s almost certainly a waste of time, and at least for k=37, there is likely no finite covering set.
    Message 1 of 2 , Sep 2, 2003
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      > Let k be a positive odd number and define P(k) to be the least value of
      > n such that 3*k*2^n-1 and 3*k*2^n+1 are both prime.
      >
      > e.g. P(49)=44 since 3*49*2^44+-1 are both prime.
      >
      > I've been looking at small values of k (<100) and got stuck with (so
      > far) 37, 41 and 51. I know that for some values of k there are no
      > solutions e.g. k=79 has a covering set of (5,7,13,17,241), but is there
      > a good way of at least looking to see whether k=37,41,51 etc. might
      > actually have covering sets and thus make looking for P(k) in these
      > cases a waste of time?

      Yes, it's almost certainly a waste of time, and at least for k=37, there
      is likely no finite covering set.

      As I posted to this list on 18 November '02:

      > The interesting value of k is k=111, which is divisible by 3.
      >
      > The probability that there exist twin primes of the form
      > 111*2^n+/-1 is vanishingly small, but apparently there is
      > no finite covering set... If such twin primes do exist,
      > it is easy to show that n == 11 (mod 36), but beyond that,
      > it gets tough.
      >
      > So k=111 is almost certainly the answer to the twin-prime form of
      > the Sierpinski problem, but may prove exceedingly difficult to
      > prove... :-(
      >
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