Can you provide a proof or counterexample for the following claim?

Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $ 4k<2^n$ , $p$ is a prime number and $n\ge3$ . Let $a$ be a natural number greater than two such that $\left(\frac{a-2}{N}\right)=-1$ and $\left(\frac{a+2}{N}\right)=-1$ where $\left(\frac{}{}\right)$ denotes Jacobi symbol. Let $S_i=P_p(S_{i-1})$ with $S_0$ equal to the modular $P_{kp^2}(a)\phantom{5} \text{mod} \phantom{5} N$. Then $N$ is prime if and only if $S_{n-2} \equiv 0 \pmod{N}$ .

You can run this test here.

I have verified this claim for $k \in [1,500]$ with $p \leq 97$ and $n \in [3,50]$ .

Further generalization of the claim

Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= 2kp^{n}\pm 1 $ where $k$ is a positive natural number , $ 2k<2^n$ , $p$ is a prime number and $n\ge3$ . Let $a$ be a natural number greater than two such that $\left(\frac{2-a}{N}\right)=\left(\frac{a+2}{N}\right)=-1$ where $\left(\frac{}{}\right)$ denotes Jacobi symbol. Let $S_i=P_p(S_{i-1})$ with $S_0$ equal to the modular $P_{kp^2}(a)\phantom{5} \text{mod} \phantom{5} N$. Then $N$ is prime if and only if $S_{n-2} \equiv -2 \pmod{N}$ .