On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers

Abstract: For an integer $k\geq 2$, let ${F^{{(k)}<em>{n}}</em>{n\geqslant} 2-k}$ be the $k$--generalized Fibonacci sequence which starts with $0, \ldots, 0,1$ (a total of $k$ terms) and for which each term afterwards is the sum of the $k$ preceding terms. In this paper, for an integer $d\geq 2$ which is square free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{{2}-dy^{2}} =\pm 1$ which is a $k$--generalized Fibonacci number, with a couple of parametric exceptions which we completely characterise. This paper extends previous work from [17] for the case $k=2$ and [16] for the case $k=3$.