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Erdos-Mollin-Walsh

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  • Paul Jobling
    Hi all, On Ribenboim p341 it says that the conjecture that there do not exists three consecutive powerful numbers is equivalent to the following (Mollin and
    Message 1 of 3 , May 3, 2001
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      Hi all,

      On Ribenboim p341 it says that the conjecture that there do not exists three
      consecutive powerful numbers is equivalent to the following (Mollin and Walsh,
      1986):

      "If m is a square-free integer, m = 7 (mod 8), if t[1]+u[1]sqrt(m) is the
      fundamental unit of the quadratic field Q(sqrt(m)), writing t[k]+u[k]sqrt(m) =
      (t[1]+u[1]sqrt(m))^k for every integer k>=1, if there exists k odd such that
      t[k] is even and powerful, then either u[k] is odd or m does not divide u[k]"

      It goes on to say that Mollin and Walsh took m=7, and t[1]+u[1]sqrt(m) as
      8+3sqrt(7), and tested up to K~114 million without finding an exception.

      But as far as I can tell with those values of t[k] and u[k], u[k] is always
      odd for odd k (u[1] is odd. u[k+2] = 48.t[k]+127.u[k], which is odd if u[k] is
      odd. QED). So the conditions of the statement are always fullfilled, without
      the need for a computer search.

      So either I am doing something wrong somewhere, or Ribenboim has misstated
      M&W. Or everybody has missed something obvious, which I doubt.

      [NB I can tie this in to primes by pointing out that if there do not exist
      three consecutive powerful numbers then there are an infinite number of
      Wieferich primes :-)]

      Regards,

      Paul.


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    • Chris Caldwell
      I am leaving now for two weeks, so if you wonder why I don t reply of fix various Prime Page stuff--it will be because I m gone. This does not effect most of
      Message 2 of 3 , Jun 13, 2001
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        I am leaving now for two weeks, so if you wonder why I don't reply of fix
        various Prime
        Page stuff--it will be because I'm gone. This does not effect most of you
        so I apologize for
        sending this to the whole list--don't you do it!

        Sorry, Chris.
      • Chris Caldwell
        As many of you know, Rudy (Rudolph Ondrejka), who for many years kept a list of record primes of various forms using his typewriter, has passed away. We will
        Message 3 of 3 , Jul 2, 2001
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          As many of you know, Rudy (Rudolph Ondrejka), who for many years kept a
          list of record primes
          of various forms using his typewriter, has passed away. We will miss
          him. I enjoyed his typed
          letters--with the texture that came from the old style typewriters striking
          the page.

          Harvey Dubner's wife has translated Rudy's list to TeX for a couple years
          now, and would continue,
          if someone would like to take over Rudy's roll of archiving these
          primes. There is some overlap
          with the Top20 pages I maintain, but not to much, take a look:

          http://www.utm.edu/research/primes/lists/top_ten/

          Harvey Dubner writes:
          >Very Sad. I have attached the last TopTen from Rudy. dvi and pdf files.
          >Do you think someone would like to continue this endeavor? My wife would
          >still be happy to keep updating the files if someone would tell her what to
          >update.

          Chris.
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