Can you provide proofs or counterexamples for the claims given below?

Inspired by Lucas-Lehmer-Riesel primality test I have formulated the following two claims:

*First claim*

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

You can run this test here .

*Second claim*

Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= k \cdot b^{n}+1 $ where $k$ is positive natural number , $ k<2^n$ , $b$ is an even positive natural 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 $S_i=P_b(S_{i-1})$ with $S_0$ equal to the modular $P_{kb/2}(P_{b/2}(a))\phantom{5} \text{mod} \phantom{5} N$. Then $N$ is prime if and only if $S_{n-2} \equiv 0 \pmod{N}$ .

You can run this test here .

I have tested these claims for many random values of $k$, $b$ and $n$ and there were no countereamples.

**EDIT**

It is possible to reformulate these claims into more compact form:

Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= k \cdot b^{n}\pm 1 $ where $k$ is positive natural number , $ k<2^n$ , $b$ is an even positive natural 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 $S_i=P_b(S_{i-1})$ with $S_0$ equal to the modular $P_{kb/2}(P_{b/2}(a))\phantom{5} \text{mod} \phantom{5} N$. Then $N$ is prime if and only if $S_{n-2} \equiv 0 \pmod{N}$ .