Perfect powers among Jacobsthal numbers
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910271992233984 |
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| author | Virbalas, Paulius |
| author_facet | Virbalas, Paulius |
| contents | Jacobsthal numbers are an example of a Lucas sequence defined by a recurrence relation analogous to that of the Fibonacci numbers, but with different parameters. In this paper, we prove that the only perfect powers among Jacobsthal numbers are the trivial ones, namely $0$ and $1$. Using the Binet formula, the problem is reduced to an exponential Diophantine equation in three unknowns. We resolve this equation via the modular approach, following the framework developed by Bennett and Skinner for ternary Diophantine equations. This work contributes to the study of perfect powers in linear recurrence sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31114 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perfect powers among Jacobsthal numbers Virbalas, Paulius Number Theory 11B39, 11D61, 11D41 Jacobsthal numbers are an example of a Lucas sequence defined by a recurrence relation analogous to that of the Fibonacci numbers, but with different parameters. In this paper, we prove that the only perfect powers among Jacobsthal numbers are the trivial ones, namely $0$ and $1$. Using the Binet formula, the problem is reduced to an exponential Diophantine equation in three unknowns. We resolve this equation via the modular approach, following the framework developed by Bennett and Skinner for ternary Diophantine equations. This work contributes to the study of perfect powers in linear recurrence sequences. |
| title | Perfect powers among Jacobsthal numbers |
| topic | Number Theory 11B39, 11D61, 11D41 |
| url | https://arxiv.org/abs/2605.31114 |