On a problem of Pillai involving S-units and Lucas numbers
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909071166144512 |
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| author | Batte, Herbert Ddamulira, Mahadi Kasozi, Juma Luca, Florian |
| author_facet | Batte, Herbert Ddamulira, Mahadi Kasozi, Juma Luca, Florian |
| contents | Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $L_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in \mathbb{N}$ independently. In the cases that $c\in \mathbb{N}$ and $c\in -\mathbb{N}$, we find all integers $c$ such that the Diophantine equation has at least three solutions. These cases are treated independently since we employ quite different techniques in proving the two cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06555 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a problem of Pillai involving S-units and Lucas numbers Batte, Herbert Ddamulira, Mahadi Kasozi, Juma Luca, Florian Number Theory 11B39, 11D61, 11D45, 11Y50 Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $L_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in \mathbb{N}$ independently. In the cases that $c\in \mathbb{N}$ and $c\in -\mathbb{N}$, we find all integers $c$ such that the Diophantine equation has at least three solutions. These cases are treated independently since we employ quite different techniques in proving the two cases. |
| title | On a problem of Pillai involving S-units and Lucas numbers |
| topic | Number Theory 11B39, 11D61, 11D45, 11Y50 |
| url | https://arxiv.org/abs/2401.06555 |