I'm studying about Pell number and Pell-Lucas number whose have Binet formula $P_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ and $Q_n=\alpha^n+\beta^n$, where $\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}$, respectively.
The simple relation between two of those is $Q^2_n-8P_n^2=4(-1)^n$.
Notice that if $q$ is odd prime factor of $P_n$ and $n$ is odd, we have $Q^2_n \equiv -4 \mod q$. "Because $q$ is odd, then $q \equiv 1 \mod 4$".
I have no idea why $q \equiv 1 \mod 4$. Does it able to be $3\mod 4$?
Thank you for attention.
Reference: Pell numbers with the Lehmer property, Bernadette Faye and Florian Luca. Afr. Mat. 2017 28:291-294