Can you prove or disprove the following claim:

>Let $N=4p+1$ where $p$ is an odd prime number , let $T_n(x)$ be the nth Chebyshev polynomial of the first kind and let $F_n(x)$ denote an irreducible factor of degree $\varphi(n)$ of $T_n(x)$ . If there exists an integer $a$ such that $F_{p}(a) \equiv 0 \pmod{N} $ then $N$ is a prime. 

You can run this test [here][1]. An incomplete  list of primes $p$ such that $4p+1$ is prime can be found [here][2] . I have verified this claim for $p \in [3,30000)$ with $a \in [1,100]$ . 


  [1]: https://sagecell.sagemath.org/?z=eJxFjE0KgzAQRvc5xeAqiVPqWHfDHKEnCBZSqiAlP6gLpbRnb3DR7j4e33tZLqyidDbXxGoLftURvQEB11qPcCIGQmh6e00PTdjZWJNhFfZpyfMUhuMur7DrLRfrV7hF6wjp7PuP4S27tgcRaPitxjRrLy2mhNOo_6FcLCwzrro6SGX4Pg_-aQx_AUx8MUg=&lang=gp&interacts=eJyLjgUAARUAuQ==
  [2]: https://oeis.org/A023212