On $${\pmb k}$$ k -Fibonacci numbers expressible as product of two Balancing or Lucas-Balancing numbers

Salah Eddine Rihane ()
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Salah Eddine Rihane: University Center of Mila

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 339-356

Abstract: Abstract The Balancing number n and the balancer r are solution of the Diophantine equation $$1+2+\cdots +(n-1) = (n+1)+(n+2)+\cdots +(n+r)$$ 1 + 2 + ⋯ + ( n - 1 ) = ( n + 1 ) + ( n + 2 ) + ⋯ + ( n + r ) . It is well known that if n is balancing number, then $$8n^2 + 1$$ 8 n 2 + 1 is a perfect square and its positive square root is called a Lucas-Balancing number. Let $$k\ge 2$$ k ≥ 2 . A generalization of the well-known Fibonacci sequence is the k-Fibonacci sequences. For these sequence the first k terms are $$0,\ldots ,0,1$$ 0 , … , 0 , 1 and each term afterwards is the sum of the preceding k terms. In this manuscript, our main objective is to find all k-Fibonacci numbers which are the product of two Balancing or Lucas-Balancing numbers.

Keywords: k-Fibonacci numbers; Balancing numbers; Lucas-Balancing numbers; Linear form in logarithms; Reduction method; 11B39; 11J86. (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00485-0

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