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Auteurs principaux: Liu, Chang, He, Bo
Format: Preprint
Publié: 2025
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Accès en ligne:https://arxiv.org/abs/2512.00548
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author Liu, Chang
He, Bo
author_facet Liu, Chang
He, Bo
contents In this paper, we consider the exponential Diophantine equation \( (2^k-1)(b^k-1)=y^q \) with $k\ge 2$, odd integer $b$ and an odd prime exponent $q$ and obtain effective upper bounds for $q$ in terms of $b$. In particular, we show that $q\le \log_2(b+1)$ holds apart from a finite, explicitly determined set of exceptional pairs $(b,q)$ when $3\le b<10^6$. As an application, we prove that the related equation \( (2^k-1)(b^k-1)=x^n, \) has no positive integer solution $(k,x,n)$ for several specific odd values of $b$, including $b\in\{5,7,11,13,21,23,27,29\}$.
format Preprint
id arxiv_https___arxiv_org_abs_2512_00548
institution arXiv
publishDate 2025
record_format arxiv
spellingShingle The diophantine equation $(2^{k}-1)(b^{k}-1)=y^{q}$
Liu, Chang
He, Bo
Number Theory
11D61
In this paper, we consider the exponential Diophantine equation \( (2^k-1)(b^k-1)=y^q \) with $k\ge 2$, odd integer $b$ and an odd prime exponent $q$ and obtain effective upper bounds for $q$ in terms of $b$. In particular, we show that $q\le \log_2(b+1)$ holds apart from a finite, explicitly determined set of exceptional pairs $(b,q)$ when $3\le b<10^6$. As an application, we prove that the related equation \( (2^k-1)(b^k-1)=x^n, \) has no positive integer solution $(k,x,n)$ for several specific odd values of $b$, including $b\in\{5,7,11,13,21,23,27,29\}$.
title The diophantine equation $(2^{k}-1)(b^{k}-1)=y^{q}$
topic Number Theory
11D61
url https://arxiv.org/abs/2512.00548