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Can you help me prove this theorem regarding Fibonacci and Lucas numbers?

Theorem.

Let

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!

Theorem.

Let

*m*,*r*ϵ Z and*n*be non-zero integer. Then

UU

_{2mn}_{+r}≡ (-1)^{mn}*U*(mod_{r}*U*) and_{m}

VV

_{2mn}_{+r}≡ (-1)^{mn}*V*(mod_{r}*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!