property of roots 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

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