How to prove following statement :
Let $~p~$ be Woodall prime of the form : $p=n\cdot 2^n-1$
$5~$ is a primitive root modulo $~p~$ iff $~n \equiv 3,6,9 \pmod {20}$
For example : $5$ is a primitive root modulo $p$ for following Woodall primes :
$3\cdot 2^3-1~,~6 \cdot 2^6-1~,~123 \cdot 2^{123}-1~,249 \cdot 2^{249}-1 ,....$
Note that : if $~n \equiv 3 ,6 ,9 \pmod {20}~$ then $~p \equiv 2,3 \pmod 5$
Euler's totient function of $~p~$ is given by :
$\phi(p)=2(n \cdot 2^{n-1}-1)$
Suppose that :
$ord_p(5) =n \cdot 2^{n-1}-1~$ where $~n \cdot 2^{n-1}-1~$ is a prime number , then :
$5^{n \cdot 2^{n-1}-1}=5^{\frac{p-1}{2}} \equiv 1 \pmod p$
and we know that :
$\left(\frac{5}{p}\right) \equiv 5^{\frac{p-1}{2}} \pmod p~$, hence :
$\left(\frac{5}{p}\right)=1 \Rightarrow p \equiv 1,4 \pmod 5$
which is a contradiction since $~p \equiv 2, 3 \pmod 5~$ , hence :
$ord_p(5) \neq n\cdot 2^{n-1}-1 \Rightarrow ord_p(5)=2(n\cdot 2^{n-1}-1)~$ , since it cannot be $~2~$ .
Therefore :
$5~$ is a primitive root modulo $~p~$ whenever $~n\cdot 2^{n-1}-1~$ is a prime number .
Any idea how to prove that $~5~$ is a primitive root modulo $~p~$ in a case when $~n\cdot 2^{n-1}-1~$ is a composite number ?