Pseudoprimality Hypothesis for Specific Class of Generalized Fermat Numbers

primus · 2015-03-20, 15:09

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 .

gd_barnes · 2015-03-25, 22:18

Quote:

Originally Posted by primus

List of generalized Fermat primes sorted by base .

http://www.noprimeleftbehind.net/crus/GFN-primes.htm

2015-03-20, 15:09	#1
primus Jul 2014 Montenegro 1A₁₆ Posts	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(2polchebyshev(b/2,1,polchebyshev(b/2,1,4)),b^2^n+1)); for(i=1,2^n-2, s=2polchebyshev(b,1,s/2)); s==0 } You can run this code here . List of generalized Fermat primes sorted by base .

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