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arXiv · 2606.18500

On the Diophantine Inequality $\lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a,b\}$

Banu İrez Aydın·Herbert Batte·İlker İnam·Florian Luca·Zeynep Demirkol Özkaya
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Abstract

In this paper, we show that there are $57$ nonnegative integer solutions $(a,b,x)$ to the inequality $1\le \lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a, b\}$ and we list them explicitly. The inequality is converted into a statement about how closely $x/q$ approximates irrational number $\sqrt{d}$ for $d\in\{2,3,6\}$, where $q$ is an integer which is $3$-smooth, after which Worley's theorem on rational approximations via continued fractions is applied to parametrise the solutions and a $p$-adic lower bound for a linear form in logarithms due to Bugeaud and Laurent is applied to find a rather large bound on $\max\{a,b\}$. We finish with an application of the LLL algorithm to reduce this bound.

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Banu İrez Aydın, Herbert Batte, İlker İnam, Florian Luca, Zeynep Demirkol Özkaya. 2026-06-16. On the Diophantine Inequality $\lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a,b\}$. https://arxiv.org/abs/2606.18500

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