Abstract
Let \(\gamma\) be an algebraic number of degree 2 and not a root of unity. In this note we show that there exists a prime ideal \({\mathfrak {p}}\) of \({\mathbb {Q}}(\gamma )\) satisfying \(\nu _{\mathfrak {p}}(\gamma ^n-1)\ge 1\), such that the rational prime p underlying \({\mathfrak {p}}\) grows quicker than n.
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Acknowledgements
The author thanks Professor Yuri Bilu for checking the proof, polishing the exposition and helpful discussions. The author also acknowledges support of China Scholarship Council Grant CSC202008310189.
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Hong, H. Stewart’s theorem revisited: suppressing the norm \(\pm 1\) hypothesis. Bol. Soc. Mat. Mex. 28, 60 (2022). https://doi.org/10.1007/s40590-022-00453-4
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DOI: https://doi.org/10.1007/s40590-022-00453-4