On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913614473986048 |
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| author | Batte, Herbert Luca, Florian |
| author_facet | Batte, Herbert Luca, Florian |
| contents | Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we completely solve the nonlinear Diophantine equation $\left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x=L_m^{(k)}$, in nonnegative integers $n$, $m$, $k$, $x$, with $k\ge 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers Batte, Herbert Luca, Florian Number Theory 11B39, 11D61, 11D45 Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we completely solve the nonlinear Diophantine equation $\left(L_{n+1}^{(k)}\right)^x+\left(L_{n}^{(k)}\right)^x-\left(L_{n-1}^{(k)}\right)^x=L_m^{(k)}$, in nonnegative integers $n$, $m$, $k$, $x$, with $k\ge 2$. |
| title | On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers |
| topic | Number Theory 11B39, 11D61, 11D45 |
| url | https://arxiv.org/abs/2412.12130 |