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More on consecutive multiplicatively dependent triples of integers

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arxiv 2411.12009 v2 pith:YRKQHTIR submitted 2024-11-18 math.NT

Michael A. Bennett , István Pink , Ingrid Vukusic This is my paper · ORCID
classification math.NT
keywords triplesdependentintegersmultiplicativelyapproximationargumentsauthorcase
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abstract

In this paper, we extend recent work of the third author and Ziegler on triples of integers $(a,b,c)$, with the property that each of $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ is multiplicatively dependent, completely classifying such triples in case $a=2$. Our techniques include a variety of elementary arguments together with more involved machinery from Diophantine approximation.

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  1. On the Lebesgue-Nagell equation $x^2-2 = y^p$

    math.NT 2025-07 conditional novelty 7.0 of 10

    For the equation x^2 - 2 = y^p, the authors prove the only solutions for p>911 are y = -1, and any nontrivial solution has y > 10^1000.

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