- Sep 13, 2007Group,... 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 continu-al

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

re-lationships:

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 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

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

[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,

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.

Conclusion:

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

recog-nized.

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