### Abstract

=\pm 1$ which is a product of two Lucas numbers, with a few exceptions that we completely characterize.

Original language | English |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | The Fibonacci Quarterly |

Publication status | Accepted/In press - 2 Aug 2019 |

### Fingerprint

### Keywords

- Lucas number
- Pell equation
- Linear forms in logarithms
- Baker's method

### ASJC Scopus subject areas

- Algebra and Number Theory

### Fields of Expertise

- Information, Communication & Computing

### Cite this

*The Fibonacci Quarterly*, 1-20.

**On the $x$--coordinates of Pell equations which are products of two Lucas numbers.** / Ddamulira, Mahadi.

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

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TY - JOUR

T1 - On the $x$--coordinates of Pell equations which are products of two Lucas numbers

AU - Ddamulira, Mahadi

PY - 2019/8/2

Y1 - 2019/8/2

N2 - Let $ \{L_n\}_{n\ge 0} $ be the sequence of Lucas numbers given by $ L_0=2, ~ L_1=1 $ and $ L_{n+2}=L_{n+1}+L_n $ for all $ n\ge 0 $. 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 product of two Lucas numbers, with a few exceptions that we completely characterize.

AB - Let $ \{L_n\}_{n\ge 0} $ be the sequence of Lucas numbers given by $ L_0=2, ~ L_1=1 $ and $ L_{n+2}=L_{n+1}+L_n $ for all $ n\ge 0 $. 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 product of two Lucas numbers, with a few exceptions that we completely characterize.

KW - Lucas number

KW - Pell equation

KW - Linear forms in logarithms

KW - Baker's method

UR - https://arxiv.org/abs/1906.06330

M3 - Article

SP - 1

EP - 20

JO - The Fibonacci Quarterly

JF - The Fibonacci Quarterly

SN - 0015-0517

ER -