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arXiv · 2610.30658

On the Open Problem in the 2023 Simon Marais Mathematics Competition

Anthony Pisani
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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.

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Anthony Pisani. 2026-10-22. On the Open Problem in the 2023 Simon Marais Mathematics Competition. https://arxiv.org/abs/2610.30658

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