Fractional part of (1+sqrt3)^2018

Question

What is the fractional part of $(1+\sqrt3)^{2018}$?

So far I reckon it must be related to $(1-\sqrt3)^{2018}$.

You can notice that $(1+\sqrt3)^{2018}=a+b\sqrt3$ and $(1-\sqrt3)^{2018}=a-b\sqrt3$ with $a,b\in \mathbb{Z}$.

$(1+\sqrt3)^{2018}+(1-\sqrt3)^{2018}=2a \in \mathbb{Z}$.

$(1+\sqrt3)^{2018}=2a-(1-\sqrt3)^{2018}$.

Can this be taken further?

Answer 1

You must have got hint from GEdgar's comment. From your work, we see that, letting $f = (1-\sqrt{3})^{2018}$

$$(1+\sqrt3)^{2018}=2a-f$$

Where $a \in \Bbb Z$. So required fraction part is $1-f = 1-(1-\sqrt 3)^{2018}$