Can you provide proof or counterexample for the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ . Let $N=2\cdot 3^n-1$ such that $n>2$ . Let $S_i=P_3(S_{i-1})$ with $S_0$ equal to the modular $P_6(P_{3}(4)) \phantom{5} \text{mod} \phantom{5} N$ . Then , $N$ is prime iff $S_{n-2} \equiv 2 \pmod{N}$
You can run this test here.
OEIS sequence of exponents $n$ for wich $N$ is prime
I was searching for counterexample using the following PARI/GP code :
CE231(lb,ub)=
{
for(n=lb,ub,
N=2*3^n-1;
S=Mod(2*polchebyshev(6,1,polchebyshev(3,1,2)),N);
ctr=1;
while(ctr<=n-2,
S=Mod(2*polchebyshev(3,1,S/2),N);
ctr+=1);
if(S==2 && !ispseudoprime(N),print(n)))
}
I have verified this claim for all $n$ up to $7000$ .
