By PFGW is Fermat and Lucas PRP.

Norm of a Gaussian Lucas number:-
norm(U(P,Q,n)),
where
U(P,Q,n)=(x^n-y^n)/(x-y);
with
x=(P+sqrt(P^2-4*Q))/2
y=(P-sqrt(P^2-4*Q))/2
being the roots of the quadratic equation
t^2-P*t+Q=0,
and P and Q being complex (Gaussian) integers;
and
norm(a+i*b)=a^2+b^2.

To be completed...