### Abstract

Original language | English |
---|---|

Pages (from-to) | 156-195 |

Number of pages | 40 |

Journal | Journal of Number Theory |

Volume | 207 |

DOIs | |

Publication status | Published - 27 Aug 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Fields of Expertise

- Information, Communication & Computing

### Cite this

**On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers.** / Ddamulira, Mahadi; Luca, Florian.

Research output: Contribution to journal › Article › Research › peer-review

*Journal of Number Theory*, vol. 207, pp. 156-195. https://doi.org/10.1016/j.jnt.2019.07.006

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TY - JOUR

T1 - On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers

AU - Ddamulira, Mahadi

AU - Luca, Florian

N1 - Journal of Number Theory 207 (2020) pp. 156-195

PY - 2019/8/27

Y1 - 2019/8/27

N2 - 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 $k=3$.

AB - 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 $k=3$.

KW - Pell equation

KW - Generalized Fibonacci sequence

KW - Linear forms in logarithms

KW - Baker's method

U2 - 10.1016/j.jnt.2019.07.006

DO - 10.1016/j.jnt.2019.07.006

M3 - Article

VL - 207

SP - 156

EP - 195

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -