On concatenations of two $k$-generalized Lucas numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909813232893952 |
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| author | Tumwesigye, Alex Behakanira Ddamulira, Mahadi Kaggwa, Prosper |
| author_facet | Tumwesigye, Alex Behakanira Ddamulira, Mahadi Kaggwa, Prosper |
| contents | For an integer \( k \geq 2 \), the sequence of \( k \)-generalized Lucas numbers is defined by the recurrence relation \( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \) for all \( n \geq 2 \), with initial conditions \( L_0^{(k)} = 2 \), \( L_1^{(k)} = 1 \) for all \( k \geq 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \geq 3 \). In this paper, we determine all \( k \)-generalized Lucas numbers that are concatenations of two terms of the same sequence and completely solve this problem for \( k \geq 3 \). Our approach combines nonzero lower bounds for linear forms in logarithms, reduction techniques based on the Baker--Davenport method and the LLL-algorithm, together with continued fraction analysis and computational verification using SageMath. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_24057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On concatenations of two $k$-generalized Lucas numbers Tumwesigye, Alex Behakanira Ddamulira, Mahadi Kaggwa, Prosper Number Theory 11B39, 11D61, 11D45, 11J86 For an integer \( k \geq 2 \), the sequence of \( k \)-generalized Lucas numbers is defined by the recurrence relation \( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \) for all \( n \geq 2 \), with initial conditions \( L_0^{(k)} = 2 \), \( L_1^{(k)} = 1 \) for all \( k \geq 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \geq 3 \). In this paper, we determine all \( k \)-generalized Lucas numbers that are concatenations of two terms of the same sequence and completely solve this problem for \( k \geq 3 \). Our approach combines nonzero lower bounds for linear forms in logarithms, reduction techniques based on the Baker--Davenport method and the LLL-algorithm, together with continued fraction analysis and computational verification using SageMath. |
| title | On concatenations of two $k$-generalized Lucas numbers |
| topic | Number Theory 11B39, 11D61, 11D45, 11J86 |
| url | https://arxiv.org/abs/2509.24057 |