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Re: Primes of the form 4^n-3

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  • jfoug@kdsi.net
    ... Especially the 4^720-3 number ;) Might you mean 4^7200-3 instead? Jim.
    Message 1 of 3 , May 3, 2001
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      --- In primenumbers@y..., paulunderwood@m... wrote:
      > Hi
      >
      >Paul Leyland askedme the other day about 4^n-3. As far as I know the
      > ...
      >4^n-3 not proven prime. n:
      >720
      > ...
      >In both case I tested to n=17400. You might want to check these are
      >PRP.

      Especially the 4^720-3 number ;) Might you mean 4^7200-3 instead?

      Jim.
    • Paul Leyland
      ... Indeed, as I am toying with the idea of searching for further examples and putting up a web page to coordinate the search, should anyone else be interested
      Message 2 of 3 , May 4, 2001
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        > Paul Leyland askedme the other day about 4^n-3. As far as I know the
        > biggest "ordinary prime" (whatever that is) is 4^7057-3 proven by
        > Preda Milhailescu.

        Indeed, as I am toying with the idea of searching for further examples
        and putting up a web page to coordinate the search, should anyone else
        be interested in joining in. However, pressure of other work means that
        it won't happen in the immediate future 8-(

        > I am intersted in these because I have never found a composite of
        > the form F:2^N-2^k-1 for which F|2^F-2. In the above case k=1 and
        > n=2N.
        >
        > Modular reducton of a 2N bit number is particularly easy since
        > 2^N=2+1 and hence requires just two shifts and additions. Moreover to
        > test F we could check 2^(2^N)=8 modulo F i.e take N (FFT) squarings
        > of 2, use the simple modular reduction, and finally check to see if
        > this equals 8.
        >
        > I checked an old email file to the old utm list server for back on
        > the 3rd May 1999.I wrote:
        >
        > >4^n-3 is proven prime. n:

        [Lists deleted.]

        Thanks Paul! This will help populate the web page, and also gives a few
        nice test cases to pick up really stupid bugs.



        Paul
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