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

TL;DR

This paper investigates the intersection of Pell equation solutions and $k$--generalized Fibonacci numbers, establishing uniqueness results for the $x$-coordinates, with specific exceptions, extending prior work for $k=2$ and $k=3$.

Contribution

It proves that at most one $x$-coordinate of a Pell equation is a $k$--generalized Fibonacci number, with a complete characterization of exceptions, generalizing previous cases.

Findings

At most one Pell $x$-coordinate is a $k$--generalized Fibonacci number.

Complete characterization of parametric exceptions.

Extension of previous results for $k=2$ and $k=3$.

Abstract

For an integer $k\geq 2$, let $\{F^{(k)}_{n}\}_{n\geqslant 2-k}$ 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 $k=3$.