PreprintCC BY-SA 4.0arXiv
On the Open Problem in the 2023 Simon Marais Mathematics Competition
Abstract
We prove that, for n ∈ ℤ+ not a perfect square, the only pairs of positive integers {a, b} for which there exists r ∈ ℝ such that ra + √n and rb + √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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BibTeX
@misc{pisani2026open,
title = {{On the Open Problem in the 2023 Simon Marais Mathematics Competition}},
author = {Anthony Pisani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30658},
eprint = {2610.30658},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30658},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On the Open Problem in the 2023 Simon Marais Mathematics Competition
AU - Anthony Pisani
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.30658
UR - https://arxiv.org/abs/2610.30658
AB - We prove that, for n ∈ ℤ^+ not a perfect square, the only pairs of positive integers {a, b} for which there exists r ∈ ℝ such that r^a + √n and r^b + √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.
ER -