Chebyhev polynomials and Primality Testing

Question

It is a well known Theorem that an odd positive integer $n$ is prime if and only if $T_{n}(x) \equiv x^n \pmod{n}$, where $T_{n}(x)$ is the $n^{th}$ Chebyshev polynomial of the first kind.

Do we deduce from this Theorem that $n$ is prime if and only if $T_{n}(x) \equiv x \pmod{x^{n-1} -1,n}$?

Any help with this will be appreciated.

PS. Please let me know if this is a duplicate. If so, will be happy to cancel.

Answer 1

In order to prove $\Longrightarrow$, I tried the following. If $n$ is prime then \begin{eqnarray*} T_{n}(x) &\equiv & x^n \pmod{n}, \\ & \equiv & x(x^{n-1} -1) + x ,\\ T_{n}(x) &\equiv & x \pmod{x^{n-1} -1,n}.\ \end{eqnarray*}