For which complex numbers $\alpha$ and $\beta$ is it true that $\alpha^n+\beta^n$ is always an integer?

Possibly a very straightforward question, but:

Question. For which complex numbers $\alpha$ and $\beta$ is it true that $\alpha^n+\beta^n$ is always an integer for all $n=1,2,3\ldots$?

For example, $$\alpha = \frac{1+i\sqrt{7}}{2}, \beta = \frac{1-i\sqrt{7}}{2}$$ have this relationship.

A couple of remarks. Firstly, a way of finding such $\alpha$ and $\beta$ pairs show's up in Silverman's book "The Arithmetic of Elliptic Curves." In particular:

Secondly, something similar seems to occur in connection with the Fibonacci numbers. Following this line of thought, perhaps a better question would be: for which complex numbers $\alpha$ and $\beta$ does there exist a complex number $k$ such that $$\frac{\alpha^n+\beta^n}{k}$$ is always an integer?

Solution 1:

This is certainly true if $\alpha$ and $\beta$ are conjugate quadratic integers because $\alpha^n+\beta^n$ is a symmetric function of $\alpha$ and $\beta$ and so is an integer polynomial expression in $\alpha+\beta$ and $\alpha\beta$.

Conversely, if $\alpha+\beta$ and $\alpha^2+\beta^2$ are integers, so is $2\alpha\beta$. Therefore, $\alpha$ and $\beta$ are roots of a polynomial $x^2+ax+\frac{b}{2}$ with $a,b\in\mathbb Z$, and so are definitely conjugate quadratic numbers, though perhaps not necessarily quadratic integers.

Now, by the same argument, $\alpha^2+\beta^2$ and $\alpha^4+\beta^4$ are integers implies $2\alpha^2\beta^2$ is an integer, that is, $2(\frac{b}{2})^2=\frac{b^2}{2}$ is an integer. Therefore, $b$ is even.

Bottom line: $\alpha^n+\beta^n$ is an integer for all $n$ iff $\alpha$ and $\beta$ are conjugate quadratic integers.