TITLE: Prove that $\left(\frac{3+\sqrt{17}}{2}\right)^n + \left(\frac{3-\sqrt{17}}{2}\right)^n$ is always odd for any natural $n$.
QUESTION [5 upvotes]: Prove that $$\left(\frac{3+\sqrt{17}}{2}\right)^n + \left(\frac{3-\sqrt{17}}{2}\right)^n$$
is always odd for any natural $n$.

I attempted to write the binomial expansion and sum it so the root numbers cancel out, and wanted to factorise it but didn't know how. I also attempted to use induction but was not sure how to proceed.

REPLY [3 votes]: A high-powered solution comes from looking at the expression $ 2 $-adically. Indeed, by choosing an embedding $ \mathbf Q(\sqrt{17}) \to \mathbf Q_2 $ and noting that we have a sum of the form $ \alpha^n + \beta^n $, we note that $ \alpha + \beta = 3 $ is odd. It follows that one of $ \alpha, \beta $ is odd and the other one is even in $ \mathbf Z_2 $, and thus, upon reduction modulo $ 2 $, the same is true for $ \alpha^n, \beta^n $ for any $ n \geq 1 $; and thus $ \alpha^n + \beta^n $ is odd.