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## Re: Fermat and Mersenne numbers are squarefree?

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• ... 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
Message 1 of 3 , Dec 26, 2001
--- 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?
• ... I answered my own question with a little research... From MathWorld: The conjecture that there are no powerful number triples implies that there are
Message 2 of 3 , Dec 26, 2001
--- In primenumbers@y..., "jbrennen" <jack@b...> wrote:
> 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?

I answered my own question with a little research...

From MathWorld:

"The conjecture that there are no powerful number triples implies
that there are infinitely many Wieferich primes (Granville 1986,
Vardi 1991)."

Originally conjectured by Erdos, which automatically makes it
"well-known" in my book...
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