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