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

Re: Primes in the interval (Problem)

Expand Messages
  • djbroadhurst
    ... Off-list, Sergey asked me for an example of a large error. Let p[n] = 10^1000 + 453. Then p[n-1] = 10^1000 - 1769 is the previous prime. Sergey defines Q
    Message 1 of 3 , Dec 6, 2010
    • 0 Attachment
      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > I have already explained why your error is unbounded

      Off-list, Sergey asked me for an example of a large error.

      Let p[n] = 10^1000 + 453.
      Then p[n-1] = 10^1000 - 1769 is the previous prime.
      Sergey defines Q as the number of primes
      between p[n-1]^2 and p[n]^2.
      Using the prime number theorem, we may estimate
      Q =~ (p[n]^2 - p[n-1]^2)/log(p[n]^2) =~ 9.65*10^999

      Sergey defines his "error" E by
      E = (p[n]^2 - p[n-1]^2)*prod(k=1,n,1-1/p[k]) - Q

      Using Mertens' theorem
      http://mathworld.wolfram.com/MertensTheorem.html
      we may easily estimate
      E =~ 2*exp(-Euler)*Q - Q =~ 1.19*10^999
      whose integer part has 1000 decimal digits.

      Clearly the error is unbounded, since it is
      of the same order as the n-th prime, p[n].

      I remark that Sergei has already advertised his "blogspot" in 9
      messages to this list, while making no significant contribution
      to our own discussions. I hope that we shall not receive a 10th
      advertisement for a site that seems to make no attempt to learn
      from careful correction of its obvious failures.

      David Broadhurst
    Your message has been successfully submitted and would be delivered to recipients shortly.