Prove by induction that $\left [ \left (3+\sqrt{5} \right )^{n} + \left (3-\sqrt{5} \right )^{n} \right ]$ is multiple of $2^{n}$ Prove by induction that :
$\left [ \left (3+\sqrt{5} \right )^{n} + \left (3-\sqrt{5} \right )^{n} \right ]$ is multiple of $ 2^{n}$
My proof is :
At $n=1$
$$\frac{\left ( 3+\sqrt{5} \right )+\left ( 3-\sqrt{5} \right )}{2} = \frac{6}{2} = 3 \in \mathbb{Z}$$
Assume $P(k)$ is true
$$\frac{\left ( 3+\sqrt{5} \right )^{k}+\left ( 3-\sqrt{5} \right )^{k}}{2^{k}} = m \in \mathbb{Z}$$
$$\Rightarrow \left ( 3+\sqrt{5} \right )^{k}+\left ( 3-\sqrt{5} \right)^{k} = 2^{k}.m \rightarrow (*)$$
At $n=k+1$
$$\left ( 3+\sqrt{5} \right )^{k+1}+\left ( 3-\sqrt{5} \right )^{k+1} = \left [ \left ( 3+\sqrt{5} \right )^{k}+\left ( 3-\sqrt{5} \right )^{k} \right ]\left [ \left ( 3+\sqrt{5} \right )+\left ( 3-\sqrt{5} \right ) \right ]-\left [ \left ( 3+\sqrt{5} \right )^{k}\left ( 3-\sqrt{5} \right )+\left ( 3-\sqrt{5} \right )^{k}\left ( 3+\sqrt{5} \right ) \right ]$$
from $(*) $:
$$\left ( 3+\sqrt{5} \right )^{k+1}+\left ( 3-\sqrt{5} \right )^{k+1} =2^{k}.m.6-\left ( 3+\sqrt{5} \right )\left ( 3-\sqrt{5} \right ) \left [ \left ( 3+\sqrt{5} \right )^{k-1}+\left ( 3-\sqrt{5} \right )^{k-1} \right ] $$
but i can't resume my proof .. i can't do any thing with $\left [ \left ( 3+\sqrt{5} \right )^{k-1}+\left ( 3-\sqrt{5} \right )^{k-1} \right ]$ .

An idea for you assuming the claim's true for exponents up to $\;k\;$ . Observe that I begin from the middle of your work:
$$\left ( 3+\sqrt{5} \right )^{k+1}+\left ( 3-\sqrt{5} \right )^{k+1} = \left [ \left ( 3+\sqrt{5} \right )^{k}+\left ( 3-\sqrt{5} \right )^{k} \right ]\left [ \left ( 3+\sqrt{5} \right )+\left ( 3-\sqrt{5} \right ) \right ]-$$
$$-\left [ \left ( 3+\sqrt{5} \right )^{k}\left ( 3-\sqrt{5} \right )+\left ( 3-\sqrt{5} \right )^{k}\left ( 3+\sqrt{5} \right ) \right ]=$$
$$=2^km\cdot6-(3+\sqrt5)(3-\sqrt5)\overbrace{\left[(3+\sqrt5)^{k-1}+(3-\sqrt5)^{k-1}\right]}^{=2^{k-1}\cdot r\,,\,\,r\in\Bbb Z}=$$
$$=2^{k+1}\cdot3m-(9-5)\cdot2^{k-1}\cdot r=2^{k+1}(3m+r).\;\;\;\;\;\square$$
