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arXiv · 2206.14027

Remarks on Catalan's equation over function fields

Anwesh Ray
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Abstract

Let $\ell$ be a prime number, $F$ be a global function field of characteristic $\ell$. Assume that there is a prime $P_\infty$ of degree $1$. Let $\mathcal{O}_F$ be the ring of functions in $F$ with no poles outside of $\{P_\infty\}$. We study solutions to Catalan's equation $X^m-Y^n=1$ over $\mathcal{O}_F$ and show that under certain additional conditions, there are no non-constant solutions which lie in $\mathcal{O}_F$, when $m,n>1$.

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Anwesh Ray. 2022-06-28. Remarks on Catalan's equation over function fields. https://doi.org/10.1016/j.ffa.2023.102271

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