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Can you prove or disprove the following claim:

Let $U_n(x)$ be Chebyshev polynomial of the second kind and let $a$ be a positive integer greater than one . If $p$ is a prime number such that $p>a+1$ , then $$U_p(a) \equiv 0 \pmod{p} \text{ or } U_p(a) \equiv 2a \pmod{p}$$.

You can run a test based on this claim here. I have verified this claim for all primes up to $100000$ with $a \in [2,100]$ .

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