On balancing and Lucas-balancing numbers expressible as product of two $k$-Fibonacci numbers
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| Format: | Preprint |
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2025
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| _version_ | 1866916904340291584 |
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| author | Tripathy, Bibhu Prasad Patel, Bijan Kumar |
| author_facet | Tripathy, Bibhu Prasad Patel, Bijan Kumar |
| contents | A positive integer $n$ is called a balancing number if there exists a positive integer $r$ such that $1 + 2 + \cdots + (n-1) = (n+1) + (n+2) + \cdots + (n+r)$. The corresponding value $r$ is known as the balancer of $n$. If $n$ is a balancing number, then $8n^{2}+1$ is a perfect square, and its positive square root is called a Lucas-balancing number. For any integer $k \geq 2$, let $\{F_{n}^{(k)} \}_{n \geq -(k-2)}$ denote $k$-generalized Fibonacci sequence which starts with $0, \dots ,1$($k$ terms) where each next term is the sum of the $k$ preceding terms. In this paper, we investigate all balancing and Lucas-balancing numbers that can be expressed as the product of two $k$-generalized Fibonacci numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On balancing and Lucas-balancing numbers expressible as product of two $k$-Fibonacci numbers Tripathy, Bibhu Prasad Patel, Bijan Kumar Number Theory 11B39, 11J86 A positive integer $n$ is called a balancing number if there exists a positive integer $r$ such that $1 + 2 + \cdots + (n-1) = (n+1) + (n+2) + \cdots + (n+r)$. The corresponding value $r$ is known as the balancer of $n$. If $n$ is a balancing number, then $8n^{2}+1$ is a perfect square, and its positive square root is called a Lucas-balancing number. For any integer $k \geq 2$, let $\{F_{n}^{(k)} \}_{n \geq -(k-2)}$ denote $k$-generalized Fibonacci sequence which starts with $0, \dots ,1$($k$ terms) where each next term is the sum of the $k$ preceding terms. In this paper, we investigate all balancing and Lucas-balancing numbers that can be expressed as the product of two $k$-generalized Fibonacci numbers. |
| title | On balancing and Lucas-balancing numbers expressible as product of two $k$-Fibonacci numbers |
| topic | Number Theory 11B39, 11J86 |
| url | https://arxiv.org/abs/2508.12238 |