Properties of Lucas sequence I want to prove the following properties of Lucas sequence: 


*

*$3\mid L_m \iff m\equiv 2\pmod 4$ 

*$L_k\equiv 3\pmod 4$, where $2\mid k$ and $3\nmid k$. 
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For the first property do we use induction? 
Does the second property follow from the first one? 

If you want to use induction for Property 1,  render
$L_{m+4}=L_{m+3}+L_{m+2}=(L_{m+2}+L_{m+1})+(L_{m+1}+L_{m})=(L_{m+1}+L_{m})+L_{m+1}+(L_{m+1}+L_{m})=3L_{m+1}+2L_{m}$
This forces $3|L_{m+4}\text{ iff }3|L_m$.
It also implies $3|(L_{m+4}+L_m)$.  Fibonacci numbers also show this property.
