On $k-$generalized Fibonacci Diophantine triples

Abstract

For $n$, $k\geq 2,~$the $k$-generalized Fibonacci sequence $\left\lbrace F_{n}^{(k)} \right\rbrace $ is defined by each term afterwards is the sum of the $k$ preceding terms with the initial values $0,0,\ldots ,0,1$ ($k$
terms). In
this paper, we prove that the system
\begin{eqnarray*}
ab+1 &=&F_{x}^{\left( k\right) } \\
ac+1 &=&F_{y}^{\left( k\right) } \\
bc+1 &=&F_{z}^{\left( k\right) }
\end{eqnarray*}%
has no solution for $1<a<b<c$ with $a\leq 10^{3}$ and some positive integers $x$, $y$ and $z$.