On the Open Problem in the 2023 Simon Marais Mathematics Competition

Source abstract

We prove that, for $n \in \mathbb{Z}^+$ not a perfect square, the only pairs of positive integers $\{a, b\}$ for which there exists $r \in \mathbb{R}$ such that $r^a + \sqrt{n}$ and $r^b + \sqrt{n}$ are both rational numbers are those of the form $\{a, a\}$ or $\{g, 2g\}$ (with $g$ odd). This resolves an open question presented in the 2023 Simon Marais Mathematics Competition.