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Re: [PrimeNumbers] Generalised Fermat numbers

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  • 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 1 of 2 , Apr 8, 2004
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      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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