Let $L_n$ be the $n$ th Lucas number, with $$L_0 = 2, L_1=1, L_{n+1} = L_n + L_{n-1}$$ Is it true that for every even number $n \geq 4$, if the number $L_n - 2$ has a prime divisor $p > 2$, then the number $L_{n+1} - 1$ is also divided by $p$? If not, then is the statement still correct with $n = 2018$? If not, how many number $n$ satisfy that if the number $L_n - 2$ has a prime divisor $p > 2$, then the number $L_{n+1} - 1$ is also divided by $p$?
(Sorry, English is my second language)
