On $k$-Fibonacci balancing and $k$-Fibonacci Lucas-balancing numbers

  • S.E. Rihane Institute of Science and Technology, University Center of Mila, Mila, Algeria
Keywords: $k$-generalized Fibonacci numbers, balancing numbers, Lucas-balancing numbers, linear form in logarithms, reduction method
Published online: 2021-06-30

Abstract


The balancing number $n$ and the balancer $r$ are solution of the Diophantine equation $$1+2+\cdots+(n-1) = (n+1)+(n+2)+\cdots+(n+r). $$ It is well known that if $n$ is balancing number, then $8n^2 + 1$ is a perfect square and its positive square root is called a Lucas-balancing number. For an integer $k\geq 2$, let $(F_n^{(k)})_n$ be the $k$-generalized Fibonacci sequence which starts with $0,\ldots,0,1,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. The purpose of this paper is to show that 1, 6930 are the only balancing numbers and 1, 3 are the only Lucas-balancing numbers which are a term of $k$-generalized Fibonacci sequence. This generalizes the result from [Fibonacci Quart. 2004, 42 (4), 330-340].

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How to Cite
(1)
Rihane S. On $k$-Fibonacci Balancing and $k$-Fibonacci Lucas-Balancing Numbers. Carpathian Math. Publ. 2021, 13 (1), 259-271.