## Fermat Primes (finite number of them)

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• to anyone, the number of Fermat primes is finite... a subjective conclusion. Fermat primes must satisfy two modulo conditions, simultaneously, to exist.
Message 1 of 1 , Jan 28, 2008
to anyone,

the number of Fermat 'primes' is finite... a subjective conclusion.

Fermat 'primes' must satisfy two 'modulo' conditions, simultaneously,
to exist.

first, i conjecture that if a Fermat number is to be consider-
ed 'prime', then F(x+1) must be 'congruent' to the following 'modulo'
condition:

i. [F(x+1)] == 2^q (mod (2^x +1)) {this equation determines the prim-
ality of the Fermat numbers ONLY when used in conjunction with the
next condition.}

second, i have 'derived' the next equation from the first condition
to characterize what "G. H. Hardy and E. M. Wright have conjectured
about the 'pair-wise'ness of Fermat 'primes'."

if F(odd) implies that [n= 1], and F(even) implies that [n= 0] when
calculated using consecutive values of 'x', then 'congruent' Fermat
numbers will be 'pair-wise' and also 'prime' after satisfying
this 'modulo' condition:

ii. 2^(x+1) +1 == (2n+1) (mod (x+1)) {this is condition i., exponent-
ially reduced to allow me to search for their 'pair-wise'ness.}.

for example,...

if x= 2...
I. [257 = F(3)] == 2^1 mod 5, or q= 1;
II. [9 == (2n+1) mod 3] implies that [n= 1];

and, x= 3...
I. [65537 = F(4)] == 2^3 mod 9, q= 3;
II. [17 == (2n+1) mod 4] implies that [n= 0].

the pattern is that simple!

a set of Fermat 'primes' must be 'congruent'(base-two) to F(x+1) & be
consecutively 'pair-wise'. Only the Fermat 'primes' retain these two
characteristics produced by both 'modulo' conditions, simultaneously.

i've evaluated both 'modulo' conditions up to x=8090 using GNU Bignum
multiple-precision arithmetic using the {try GMP!} interpreter, but
could not calculate F(131071) modulo... something, to allow me to
deny the next possible Fermat 'prime' set.

the number of Fermat 'primes' is finite... as far as I can calculate
them.

regards, Bill B.
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