Question 113
Find all positive integers $m$ such that
$$
\frac{2^m \alpha^m - (\alpha + \beta)^m - (\alpha - \beta)^m}{3\alpha^2 + \beta^2}
$$
is an integer for all integer values of $\alpha, \beta$ with $\alpha\beta \neq 0$.
$$
\frac{2^m \alpha^m - (\alpha + \beta)^m - (\alpha - \beta)^m}{3\alpha^2 + \beta^2}
$$
is an integer for all integer values of $\alpha, \beta$ with $\alpha\beta \neq 0$.