- Sep 16, 2007edited 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

should say... has been once removed, not... preserved.

> 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 be [n=1]... 'n' hasn't changed... it will occur

> First, F(0)= 3 cannot be tested using H(x) since the

> modulo portion of the equation doesnt 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,

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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