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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.00548 |
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| _version_ | 1866914457043599360 |
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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 |