## 19091finite essay2

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• Sep 16, 2007
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edited from earlier e-mail... no comments? the
'pair-wise effort has to be continuous to produce more
Fer-mat primes.

> Group,... know that this is a modest effort to
> explain why only so few Fermat numbers are 'prime'.
>
> Overview:
> It can be shown that F(1)...F(4) are the only Fermat
> numbers that can establish and maintain a continual
> pair-wise condition; these Fermat numbers are all
> prime numbers, and F(0) cannot be recognized using
> the pair-wise condition.
>
> Demonstration:
> 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
> relationships:
>
> 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.

should say... has been once removed, not... 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 ex-
> pression is to scale, and 2 == 2n is a valid com-
> parison without the need for modular reduction;
> if x=1, then H(1)= 5 == (2n+1) (mod 2) suggests that
> [n=0].
>
> 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,

should be [n=1]... 'n' hasn't changed... it will occur
for the next value of 'x' since the exponential
argu-ment has been once removed.
however, G(4) has revealed that 2^q = 62 and 'q' isn't
an /odd whole number/

> 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 con-
> tinuity for an odd x have also been lost as we
> find that F(6) tests as a composite number.
>
> Conclusion:
> If G. H. Hardy and E. M. Wright were correct to in-
> sist 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 can retain this con-
> tinuity. F(0) is a small prime number also defin-
> ed by the Fermat number formula, but a pair-wise
> condition cannot be recognized.
>
>

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