Inspired by Theorem 5 in this paper I have formulated the following claim :
Let $n$ be an odd number and $n>1$ . Let $T_n(x)$ be Chebyshev polynomial of the first kind and let $P_n(x)$ be Legendre polynomial , then $n$ is a prime number if and only if the following congruences hold simultaneously
$\bullet \: T_n(3) \equiv 3 \pmod n$
$\bullet \: P_n(3) \equiv 3 \pmod n$
You can run this test here .
I was searching for pseudoprimes using the following PARI/GP program :
CL(lb,ub)=
{
forstep(n=lb,ub,[2],
if(!ispseudoprime(n),
if((Mod(polchebyshev(n,1,3),n)==3),
if((Mod(pollegendre(n,3),n)==3),print(n)))))
}
I have tested this claim up to $1.4 \cdot 10^6$ and there were no counterexamples .
Question : Can you provide a proof or a counterexample for the claim given above ?