## Abstract

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.

Original language | English |
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Pages (from-to) | 156-195 |

Number of pages | 40 |

Journal | Journal of Number Theory |

Volume | 207 |

Early online date | 27 Aug 2019 |

DOIs | |

Publication status | Published - 1 Feb 2020 |

## Keywords

- Pell equation
- Generalized Fibonacci sequence
- Linear forms in logarithms
- Baker's method
- Reduction method
- Linear form in logarithms

## ASJC Scopus subject areas

- Algebra and Number Theory

## Fields of Expertise

- Information, Communication & Computing