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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