Loading ...
Sorry, an error occurred while loading the content.

limit = 6?

Expand Messages
  • Werner D. Sand
    Let p
    Message 1 of 3 , Aug 3, 2007
    • 0 Attachment
      Let p < q be consecutive prime numbers.
      Does sum(1/(q-p*ln q), p=2..inf, converge exactly to 6?
    • Adam
      Does it even converge? If gap size at p is about ln(p) then (with all equalities being approximately equal to ) q=p+ln(p) so q-p*ln(q)=p+ln(p)-p*ln(p+ln (p))
      Message 2 of 3 , Aug 8, 2007
      • 0 Attachment
        Does it even converge?

        If gap size at p is about ln(p) then (with all equalities
        being "approximately equal to") q=p+ln(p) so q-p*ln(q)=p+ln(p)-p*ln(p+ln
        (p)) is on the order of p*ln(p) and the improper integral of 1/[x*ln
        (x)] diverges.

        It should it diverge to -oo about as fast as ln(ln(p)).

        Adam

        >
        > Let p < q be consecutive prime numbers.
        > Does sum(1/(q-p*ln q), p=2..inf, converge exactly to 6?
        >
      • Werner D. Sand
        That s right. One could also say: q ~ p, for p/q ~ 1. Then 1/(q-p*ln q) ~ 1/(p-p*ln p) and int(1/(x-x*ln x)) = -ln (ln x - 1) - -inf. Thanks. Can you
        Message 3 of 3 , Aug 9, 2007
        • 0 Attachment
          That's right. One could also say: q ~ p, for p/q ~ 1. Then 1/(q-p*ln
          q) ~ 1/(p-p*ln p) and int(1/(x-x*ln x)) = -ln (ln x - 1) -> -inf.
          Thanks. Can you nevertheless find out p when sum(1/(q-p*ln(q)) < 6
          for the first time?

          Werner



          --- In primenumbers@yahoogroups.com, "Adam" <a_math_guy@...> wrote:
          >
          > Does it even converge?
          >
          > If gap size at p is about ln(p) then (with all equalities
          > being "approximately equal to") q=p+ln(p) so q-p*ln(q)=p+ln(p)-p*ln
          (p+ln
          > (p)) is on the order of p*ln(p) and the improper integral of 1/[x*ln
          > (x)] diverges.
          >
          > It should it diverge to -oo about as fast as ln(ln(p)).
          >
          > Adam
          >
          > >
          > > Let p < q be consecutive prime numbers.
          > > Does sum(1/(q-p*ln q), p=2..inf, converge exactly to 6?
          > >
          >
        Your message has been successfully submitted and would be delivered to recipients shortly.