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Generalised Fermat numbers

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  • mad37wriggle
    I m sure this is covered somewhere, but... Let Gfp(n) = the pair (a,b) leading to the smallest prime of the form a^(2^n)+b^(2^n). Gfp(1) = (1,2) [1 digit]
    Message 1 of 2 , Apr 7 12:41 PM
      I'm sure this is covered somewhere, but...

      Let Gfp(n) = the pair (a,b) leading to the smallest prime of the form a^(2^n)+b^(2^n).

      Gfp(1) = (1,2) [1 digit]
      Gfp(2) = (1,2) [2 digits]
      Gfp(3) = (1,2) [3 digits]
      Gfp(4) = (1,2) [5 digits]
      Gfp(5) = (8,9) [31 digits]
      Gfp(6) = (8,11) [68 digits]
      Gfp(7) = (20,27) [184 digits]
      Gfp(8) = (5,14) [294 digits]
      Gfp(9) = (2,13) [571 digits]
      Gfp(10) = (26,47) [1713 digits] (PRP only)
      Gfp(11) = (3,22) [2750 digits] (PRP only)
      Gfp(12) = ? [ > 5700 digits]

      How far is Gfp(n) known?

      Also, is there an efficient primality test for numbers of this form?

      Richard
    • mikeoakes2@aol.com
      In a message dated 08/04/04 02:37:42 GMT Daylight Time, ... In August 2000 I did a complete search for primes of the form a^(2^n)+b^(2^n) for 2
      Message 2 of 2 , Apr 8 12:23 AM
        In a message dated 08/04/04 02:37:42 GMT Daylight Time,
        fitzhughrichard@... writes:


        > Let Gfp(n) = the pair (a,b) leading to the smallest prime of the form
        > a^(2^n)+b^(2^n).
        >
        > Gfp(1) = (1,2) [1 digit]
        > Gfp(2) = (1,2) [2 digits]
        > Gfp(3) = (1,2) [3 digits]
        > Gfp(4) = (1,2) [5 digits]
        > Gfp(5) = (8,9) [31 digits]
        > Gfp(6) = (8,11) [68 digits]
        > Gfp(7) = (20,27) [184 digits]
        > Gfp(8) = (5,14) [294 digits]
        > Gfp(9) = (2,13) [571 digits]
        > Gfp(10) = (26,47) [1713 digits] (PRP only)
        > Gfp(11) = (3,22) [2750 digits] (PRP only)
        > Gfp(12) = ? [ > 5700 digits]
        >
        > How far is Gfp(n) known?
        >

        In August 2000 I did a complete search for primes of the form a^(2^n)+b^(2^n)
        for 2<=n<=13 and a<b<=100, so I can confirm your results and supply the next
        2 data points.

        Gfp(12) = (2,53) [7063 digits] (PRP only)

        Gfp(13) = (43,72) [15216 digits] (PRP only)
        This is in Henri Lifchitz's "Top 5000 PRP" site
        http://ourworld.compuserve.com/homepages/hlifchitz/
        as
        <A HREF="http://www.primenumbers.net/prptop/detailprp.php?rank=951">951</A> 72^8192+43^8192 15216 Mike Oakes 08/2000

        >Also, is there an efficient primality test for numbers of this form?

        None is known.

        -Mike Oakes


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