property of roots

Question

I need to prove $\gamma_n$ is an integer for any value of $n$. $\gamma_n$ is defined as $= \alpha^n + \beta^n$. $\alpha$ and $\beta$ are roots of equation $x^2 + mx - 1 = 0$, $m$ is an odd integer.

I tried by induction, but do not know how to prove $\gamma_n$ is an integer for $n + 1$. Can some body help

Answer 1

$\alpha$ and $\beta$ are algebraic integers, hence so is $\alpha^n+\beta^n$ for each $n$. Moreover, $\alpha^n+\beta^n$ is fixed by $\mathrm{Gal}(\mathbb{Q}(\alpha)/\mathbb{Q})$, so is a rational number. And if a rational number is integral over $\mathbb{Z}$, then it's an integer.

Answer 2

We do the induction step, using strong induction. Suppose that $\gamma_k$ is an integer for all $k\lt n$. We show that $\gamma_n$ is an integer. Note that $$\alpha^n+\beta^n=(\alpha^{n-1}+\beta^{n-1})(\alpha+\beta)-\alpha\beta(\alpha^{n-2}+\beta^{n-2}).$$ But $\alpha+\beta$ and $\alpha\beta$ are integers, and the result follows.

Answer 3

$\gamma_{n+1} = \alpha^{n+1} + \beta^{n+1}= (\alpha^n+\beta^n)(\alpha+\beta)- \alpha\beta(\alpha^{n-1}+\beta^{n-1})= -m\gamma_n+\gamma_{n-1}$. You can use this identity to proceed at the inductive step.