## Erdos-Mollin-Walsh

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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
Message 1 of 3 , May 3 7:42 AM
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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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• 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.
• 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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