Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909457099784192 |
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| author | Alvarenga, Roberto Chaves, Ana Paula Ramos, Maria Eduarda Silva, Matheus Sosa, Marcos |
| author_facet | Alvarenga, Roberto Chaves, Ana Paula Ramos, Maria Eduarda Silva, Matheus Sosa, Marcos |
| contents | Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = F_{2n+1}. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the Diophantine equation (F_{n}^{(k)})^2 + (F_{n+d}^{(k)})^2 = F_{m}^{(k)} over the k-generalized Fibonacci numbers for every k \geq 2, generalizing Chaves and Marques. Next, we solve F_{n}^{s} + F_{n+d}^{s} = F_m over the Fibonacci numbers for every s \geq 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_{n}^s + \cdots + F_{n+d}^s = F_m for d+1 < n and s \geq 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers Alvarenga, Roberto Chaves, Ana Paula Ramos, Maria Eduarda Silva, Matheus Sosa, Marcos Number Theory Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = F_{2n+1}. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the Diophantine equation (F_{n}^{(k)})^2 + (F_{n+d}^{(k)})^2 = F_{m}^{(k)} over the k-generalized Fibonacci numbers for every k \geq 2, generalizing Chaves and Marques. Next, we solve F_{n}^{s} + F_{n+d}^{s} = F_m over the Fibonacci numbers for every s \geq 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_{n}^s + \cdots + F_{n+d}^s = F_m for d+1 < n and s \geq 2. |
| title | Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.08899 |