On the finiteness of solutions for certain Diophantine equations

The Ramanujan Journal Pub Date : 2024-07-10 DOI:10.1007/s11139-024-00897-4

Mohamed Ouzahra

引用次数: 0

Abstract

We study Diophantine equations of the form \( f(n)=m^2 \pm a,\; n, m\in \mathbb {N}, \)where \(a\in \mathbb {N}^*\) and \(f: \mathbb {N} \rightarrow \mathbb {N}\) tends to \(+\infty . \) Necessary and sufficient conditions for the set of solutions to be finite are formulated in terms of asymptotic properties and the repartition of the digits of the fractional part of \(\sqrt{f(n)}\)

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论某些 Diophantine 方程解的有限性

我们研究了形式为（ f(n)=m^2 \pm a,\; n, min \mathbb {N}, \）的二叉方程，其中（ ain \mathbb {N}^*\) 和（ f: \mathbb {N} \rightarrow \mathbb {N}\ ）趋向于（ +\infty .\从渐近性质和 \(\sqrt{f(n)}\) 小数部分的数位重新划分的角度，提出了解集是有限的必要条件和充分条件。）

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The Ramanujan Journal

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