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Twins

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  • mad37wriggle
    If there is a web page somewhere dealing with exactly this then I would appreciate it if someone would point me in its direction... Let k be a positive odd
    Message 1 of 2 , Sep 2, 2003
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      If there is a web page somewhere dealing with exactly this then I would
      appreciate it if someone would point me in its direction...

      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?

      Thanks,

      Richard
    • 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 2 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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