- 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,
"If m is a square-free integer, m = 7 (mod 8), if t+usqrt(m) is the
fundamental unit of the quadratic field Q(sqrt(m)), writing t[k]+u[k]sqrt(m) =
(t+usqrt(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+usqrt(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 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 :-)]
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- I am leaving now for two weeks, so if you wonder why I don't reply of fix
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!
- 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
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:
Harvey Dubner writes:
>Very Sad. I have attached the last TopTen from Rudy. dvi and pdf files.Chris.
>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