On Wagstaff primes in the $k$-Lucas number sequence
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915813035868160 |
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| author | Batte, Herbert |
| author_facet | Batte, Herbert |
| contents | A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)/3$, where $\mathfrak{p}$ is an odd prime. Let $(L_n^{(k)})_{n\geq 2-k}$ be the $k$-Lucas number sequence defined by the recurrence relation
$ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)}$, for all $n \ge 2$, with initial terms \( L_0^{(k)} = 2 \) and \( L_1^{(k)} = 1 \) for all \( k \ge 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \ge 3 \). In this paper, we show that the only solutions to the Diophantine equation $L_n^{(k)} = (2^{\mathfrak{p}}+1)/3$ are $(n,k,\mathfrak{p})\in\{(5,2,5),(6,4,7)\}\cup \{(2,k,3):k\ge 2\}$. We use linear forms in logarithms and the LLL reduction method to prove our result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20387 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Wagstaff primes in the $k$-Lucas number sequence Batte, Herbert Number Theory 11A41, 11B39, 11D61, 11D45 A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)/3$, where $\mathfrak{p}$ is an odd prime. Let $(L_n^{(k)})_{n\geq 2-k}$ be the $k$-Lucas number sequence defined by the recurrence relation $ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)}$, for all $n \ge 2$, with initial terms \( L_0^{(k)} = 2 \) and \( L_1^{(k)} = 1 \) for all \( k \ge 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \ge 3 \). In this paper, we show that the only solutions to the Diophantine equation $L_n^{(k)} = (2^{\mathfrak{p}}+1)/3$ are $(n,k,\mathfrak{p})\in\{(5,2,5),(6,4,7)\}\cup \{(2,k,3):k\ge 2\}$. We use linear forms in logarithms and the LLL reduction method to prove our result. |
| title | On Wagstaff primes in the $k$-Lucas number sequence |
| topic | Number Theory 11A41, 11B39, 11D61, 11D45 |
| url | https://arxiv.org/abs/2602.20387 |