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Old 2015-03-20, 15:09   #1
primus
 
Jul 2014
Montenegro

2·13 Posts
Default Pseudoprimality Hypothesis for Specific Class of Generalized Fermat Numbers

Definition

Let P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right) , where m and x are nonnegative integers .

Conjecture

Let F_n(b)=b^{2^n}+1 such that n>1 , b is even , 3 \not\mid b and 5\not\mid b .

Let S_i=P_b(S_{i-1}) with S_0=P_{b/2}(P_{b/2}(8)) , thus

F_n(b) is prime iff S_{2^n-2} \equiv 0 \pmod{F_n(b)}

PARI/GP implementation

Code:
GF(b,n)=
{ 
my(s=Mod(2*polchebyshev(b/2,1,polchebyshev(b/2,1,4)),b^2^n+1));
for(i=1,2^n-2, s=2*polchebyshev(b,1,s/2));
s==0 
}
You can run this code here .

List of generalized Fermat primes sorted by base .
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Old 2015-03-25, 22:18   #2
gd_barnes
 
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May 2007
Kansas; USA

287F16 Posts
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Quote:
Originally Posted by primus View Post
List of generalized Fermat primes sorted by base .
http://www.noprimeleftbehind.net/crus/GFN-primes.htm
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