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Re: Fermat and Mersenne numbers are squarefree?

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  • jbrennen
    ... 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
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      --- 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?
    • jbrennen
      ... 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
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        --- 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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