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18989Re: Thanks to all. Proposed binary Prime number test fails.

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  • elevensmooth
    Jun 5, 2007
      --- In primenumbers@yahoogroups.com, "Kevin Acres" <research@...> wrote:

      > just wondering ...
      > if it holds true that p^2 always divides the
      > minimal base^(p-1/x)-1 that p does.

      Yes. Let d be the smallest value so that p divides b^d-1.

      First observe that if b^x-1 and b^y-1 and both divisible by p^2 (or
      any other number), then b^(gcd(x,y))-1 is also divisible by p^2 (or
      any other number).

      Second, observe that b^(pd)-1 is divisible by p^2. To see this,
      express b^d as (ps+1), then expand (ps+1)^p with the binomial theorem.

      Third, observe that every case of divisibility by p is of the form
      b^(kd)-1, and every case of divisibility by p^2 is also a case of
      divisiblity by p, so the minimal case of divisibility by p^2 must be
      of the form b^(kd)-1.

      Finally, if there is any k<p such that b^(kd)-1 is divisible by p^2,
      then b^gcd(kd,pd) must also be divisible by p^2, but gcd(kd,pd)=d.

      Therefore, in all cases of Vanishing Fermat Quotients, p^2 divides the
      same primitive term that p divides.

      Poohbah of OddPerfect.org

      P.S. We still need a few large factors of Vanishing Fermat Quotients
      at http://oddperfect.org/FermatQuotients.html
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