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

For an integer k≥2, let {F _n ^(k)} _n≥2−k be the k–generalized Fibonacci sequence which starts with 0,…,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≥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=±1, which is a k–generalized Fibonacci number, with a couple of parametric exceptions which we completely characterize. This paper extends previous work from [18] for the case k=2 and [17] for the case k=3.