Properties of Lucas sequence

Question

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 $3\nmid k$.

$$$$

For the first property do we use induction?

Does the second property follow from the first one?

Answer 1

In both cases, it helps to consider the Lucas numbers modulo $m$. For example, modulo $3$, the Lucas numbers (zero-indexed) begin $$2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, \dots$$ and you may be able to spot a periodic pattern here: the sequence $2,1,0,1,1,2,0,2$ repeats over and over.

If you prove this periodic pattern, then your first statement follows just by looking where the $0$ appears. Similarly, if you find and prove a periodic pattern for $L_k \bmod 4$, then the second statement will follow by looking where the $3$ appears.

The pattern can be proved by some kind of induction. Essentially, knowing $L_k$ and $L_{k+1}$ modulo $m$ tells you $L_{k+2}$ modulo $m$, so once the mod-$m$ sequence repeats its starting values of $2,1$ once, you know that it will repeat them forever.

Answer 2

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.