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19085finite Fermat essay

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  • Bill Bouris
    Sep 13, 2007
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      Group,... know that this is a modest effort to explain
      why only so few Fermat numbers are 'prime'.

      It can be shown that F(1)...F(4) are the only Fermat
      numbers that can establish and maintain a continu-al
      ‘pair-wise’ condition; these Fermat numbers are all
      ‘prime’ numbers, and F(0) cannot be recognized using
      the ‘pair-wise’ condition.

      If a Fermat number is of the form F(x)= 2^(2^(2^x))+1
      and x is a /whole number/, then a continual
      ‘pair-wise’ condition can be supported by these modulo

      I. If x is a /whole number/, then G(x)=
      (2^(2^(2^(x+1)))+1) == 2^q (mod (2^x+1)) where q is
      /zero or an odd natural number/ which reduces to...

      II. H(x)= (2^(x+1)+1)== (2n+1) (mod (x+1)) where the
      expression (2n+1) represents the number ‘q’ from the
      previous equation, and the exponential argument has
      been neatly preserved.

      First, F(0)= 3 cannot be tested using H(x) since the
      ‘modulo’ portion of the equation doesn’t make sense;
      this Fermat number is considered to be similar to that
      of geometric point at/near infinity.

      Now, if x=0, then H(0)= 3 == (2n+1) (mod 1) suggests
      that [n=1] since a unitary modulo implies the
      expres-sion is ‘to scale’, and 2 == 2n is a valid
      comparison without the need for modular reduction; if
      x=1, then H(1)= 5 == (2n+1) (mod 2) suggests that

      The ‘pair-wise’ condition for both an even and odd ‘x’
      has been established, and a change in the value for
      ‘n’ would indicate a change in the condition.

      If x=2, then H(2)= 9 == (2n+1) (mod 3) suggests that
      [n=1], and the ‘pair-wise’ condition for an even ‘x’
      is maintained.

      If x=3, then H(3)= 17 == (2n+1) (mod 4) suggests that
      [n=0], and the ‘pair-wise’ pattern is unchanged for an
      odd ‘x’.

      However, if x=4, then H(4)= 33 == (2n+1) (mod 5)
      suggests that [n=0]. The value of ‘n’ has changed,
      and a loss of continuity is confirmed since F(5) tests
      as a ‘composite’ number.

      If x=5, then H(5)= 65 == (2n+1) (mod 6) suggests that
      [n=2]; the ‘pair-wise’ condition and its continuity
      for an odd ‘x’ have also been lost as we find that
      F(6) tests as a ‘composite’ number.

      If G. H. Hardy and E. M. Wright were correct to insist
      that a ‘pair-wise’ condition is continually linked to
      the presence of Fermat ‘prime’ numbers, then I have
      demonstrated that the F(1)...F(4) are the only Fermat
      numbers that could retain this continuity. F(0) is a
      small ‘prime’ number also defined by the Fermat number
      formula, but a ‘pair-wise’ condition cannot be

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