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  • leavemsg1
    Aug 2, 2007
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      I have one rather simple explanation for why I
      think that the number of Fermat primes is finite.

      Conjecture:

      If a Fermat number of the form F(k,m)= 2^(2^((2^k)*m)) +1
      is prime, then [2^(2^(2^k)*m) +1 == 2^q (mod 2^(2^k*m) +1)]
      where 'k' is /-1 or a whole number/ and 'k' <= 'm' which is
      a /whole number/ for some /odd number/ 'q'.

      This argument further reduces to...[2^(2^k)*m) == (2n +/-1) mod
      (2^k*m -k)] and is only satisfied when a combination of 'k' and 'm'
      can be found to confirm that n= 0. The modulo equation is nasty,
      but the results are wonderful!


      Examine the following modulo equations:

      expanded...
      k=-1, m=0; [(2^((2^(-1))*0)) == (2n+1) mod ((2^(-1))*0) -(-1)]
      =>[2^0 == (2n+1) mod (0+1)] => [1 == (2n+1) mod 1]
      => [0 == (2n) mod 1]=> [2n= 0] => [n= 0]; F0= 3 is prime,


      continued...
      k= 0, m=1; [2 == (2n-1) mod 1] => [0 == (2n) mod 1]
      => [2n= 0] => [n= 0]; F1= 5 is prime,

      k= 1, m=1; [4 == (2n+1) mod 1] => [0 == (2n) mod 1]
      => [2n= 0] => [n= 0]; F2= 17 is prime,

      k= 0, m=3; [8 == (2n-1) mod 3] => [0 == (2n) mod 3]
      => [2n= 0] => [n= 0]; F3=257 is prime,
      k= 1, m=2; [16 == (2n+1) mod 3] => [0 == (2n) mod 3]
      => [2n= 0] => [n= 0]; F4= 65537 is prime.

      but...
      k= 0, m=5; [32 == (2n-1) mod 5] => [3 == (2n) mod 5]
      => [2n= 3] => [n= fraction]; and F5 is composite!

      k= 0, m=6; [64 == (2n+1) mod 6] => [3 == (2n) mod 6] =>
      [2n= 3] => [n= fraction]; and k= 1,m=3; [64 == (2n+1) mod 5]
      => [3 == (2n) mod 5] => [2n= 3] => [n= fraction]; & F6 is composite!

      k= 0, m=7; [128 == (2n-1) mod 7]=> [3 == (2n) mod 7]
      => [2n= 3] => [n= fraction]; so F7 is composite!

      and...
      k= 0, m=10;[256 == (2n+1) mod 10]=> [5 == (2n) mod 10
      => [2n= 5]=> [n= fraction]; and k= 2,m=5; [256 == (2n+1) mod 8]
      => [7 == (2n) mod 8]=> [2n= 7] => [n= fraction]; & F10 is composite!
      etc.

      Hence, the number of Fermat primes is finite because the pattern
      ends abruptly on the 5th number in the sequence, and thereafter.

      Thanks, Bill