Representing an integer and its powers in two unrelated number systems
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917399123460096 |
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| author | Sharma, Divyum Singhal, L. |
| author_facet | Sharma, Divyum Singhal, L. |
| contents | Let $α$ be a fixed quadratic irrational. Consider the Diophantine equation \[
y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent denominators to $α$. We find two effective upper bounds for $y^a$ which depend on the Hamming weights of $y$ with respect to its radix and Zeckendorf representations, respectively. The latter bound extends a recent result of Vukusic and Ziegler. En route, we obtain an analogue of a theorem by Kebli, Kihel, Larone and Luca. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representing an integer and its powers in two unrelated number systems Sharma, Divyum Singhal, L. Number Theory 11D61, 11J86, 11A63, 11B39 Let $α$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent denominators to $α$. We find two effective upper bounds for $y^a$ which depend on the Hamming weights of $y$ with respect to its radix and Zeckendorf representations, respectively. The latter bound extends a recent result of Vukusic and Ziegler. En route, we obtain an analogue of a theorem by Kebli, Kihel, Larone and Luca. |
| title | Representing an integer and its powers in two unrelated number systems |
| topic | Number Theory 11D61, 11J86, 11A63, 11B39 |
| url | https://arxiv.org/abs/2502.00296 |