Can you help me prove this theorem regarding Fibonacci and Lucas numbers?
Theorem.
Let m,r ϵ Z and n be non-zero integer. Then
U_{2mn}_{+r} ≡ (-1)^{mn} U_{r} (mod U_{m}) and
V_{2mn}_{+r} ≡ (-1)^{mn} V_{r} (mod U_{m}).
Im not that good at proving. This type of congruence is much harder than what I read in our book, but I badly need the proof for this one, even just this one, to understand better Fiboancci and Lucas numbers.
I'd be glad to hear from you soon.
Thank you very much!