--- In primenumbers@y..., Pavlos N <pavlos199@y...> wrote:

> Can anyone comment or find a(possible) flaw in :

> http://www.dybot.com/numbers/sqfree.htm

Yes, there is a flaw. I can't pinpoint the exact error in

reasoning (basically, I don't want to spend the time to find

it :-), but the equation which is "proven" in section 4.2

is not true. There is an easily computable counterexample.

The author states that:

ß(p^n) = p^(n-1)·ß(p) (for any prime p)

This is not true if p = 1093 & n = 2:

ß(p^n) == 364

p^(n-1)·ß(p) == 1093*364

I believe that in general, the postulate in section 4.2 is

not true for any Wieferich prime.

Somebody apparently tried to prove the Mersennes squarefree

using this technique in '96 and came up against this same

problem:

http://www2.netdoor.com/~acurry/mersenne/archive2/0037.html
I seem to remember reading about another well-known number theory

conjecture which had something to do with Wieferich primes. If I

remember correctly, either the truth or falsehood of said

conjecture (I don't remember which) would imply an infinitude of

Wieferich primes. Can anyone refresh my memory on this?