Primality test for specific class of natural numbers using factors of Lucas polynomials

Question

This question is related to my previous question.

Can you prove or disprove the following claim:

Let $N=2n+1$ where $n$ is an odd natural number greater than one , let $L_m(x)$ be the mth Lucas polynomial and let $F_m(x)$ denote an irreducible factor of degree $\varphi(m)$ of $L_m(x)$ . If there exists an integer $a$ such that $F_{n}(a) \equiv 0 \pmod{N} $ then $N$ is a prime.

You can run this test here. I have verified this claim only for small values of $N$ , that is $N \in [7,1000]$ with $a \in [1,100]$ , because my PARI/GP implementation of the test is too slow.

EDIT

For values ​​of $n$ that are odd prime numbers this test runs in polynomial time (PARI/GP implementation) . List of Sophie Germain primes can be found here.

Answer 1

This claim can be proved in essentially the same way as the previous one. We have $$F_n(x)=\prod_{\substack{|m|<n/2\\(m,n)=1}}(x+\zeta^m-\zeta^{-m}),$$ where $\zeta\in\mathbb{C}$ is a primitive $2n$-th root of unity. The splitting field of $F_n(x)$ is the $n$-th cyclotomic field.

Assume that $q\nmid n$ is a prime number such that the reduction of $F_n(x)$ mod $q$ has a root in $\mathbb{F}_q$. The roots of $F_n(x)$ in $\overline{\mathbb{F}_q}$ are of the form $\xi^m-\xi^{-m}$, where $\xi\in\overline{\mathbb{F}_q}$ is a primitive $2n$-th root of unity. By assumption, the Frobenius automorphism $t\mapsto t^q$ fixes one of these roots, which is only possible when $q\equiv 1\pmod{2n}$. It follows that, for any $a\in\mathbb{Z}$, the prime factors of $F_n(a)$ coprime to $n$ are congruent to $1$ modulo $2n$. In particular, if $2n+1$ divides $F_n(a)$, then the only prime factor of $2n+1$ can be itself, i.e., $2n+1$ is prime.