# Compositeness testing using Chebyshev polynomials of the second kind

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 $$a \in [2,100]$$ .