On the sum of squares of consecutive $k$-bonacci numbers which are $l$-bonacci numbers
Volume 156 / 2019
Colloquium Mathematicum 156 (2019), 153-164
MSC: Primary 11Dxx; Secondary 11J86, 11B39.
DOI: 10.4064/cm7272-6-2018
Published online: 11 January 2019
Abstract
Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_{m+2}=F_{m+1}+F_m$, for $m\geq 0$, where $F_0=0$ and $F_1=1$. A well-known generalization of the Fibonacci sequence is the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ which is defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In 2014, Chaves and Marques solved the Diophantine equation $(F_n^{(k)})^2+(F_{n+1}^{(k)})^2=F_m^{(k)}$ in integers $m,n$ and $k\geq 2$. In this paper, we generalize this result by proving that the Diophantine equation \[ (F_n^{(k)})^2+(F_{n+1}^{(k)})^2=F_m^{(l)} \] has no solution in positive integers $n,m,k,l$ with $2 \leq k \lt l$ and $n \gt 1$.
