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Can you provide a proof or a 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 $E_n(b)= \frac{b^{2^n}+1}{2} $ where $b$ is an odd natural number greater than one and $n\ge2$ . Let $a$ be a natural number greater than two such that $\left(\frac{a-2}{E_n(b)}\right)=-1$ and $\left(\frac{a+2}{E_n(b)}\right)=-1$ where $\left(\frac{}{}\right)$ denotes Jacobi symbol. Let $S_i=P_b(S_{i-1})$ with $S_0$ equal to the modular $P_{b^2}(a)\phantom{5} \text{mod} \phantom{5} E_n(b)$. Then if $E_n(b)$ is prime then $S_{2^n-2} \equiv a \pmod{E_n(b)}$ .

You can run this test here.

I have tested this claim for many random values of $b$ and $n$ and there were no counterexamples.

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