### Abstract

Let P and Q be non-zero relatively prime integers. The Lucas sequence {U_{n}(P,Q)} is defined by U_{0}=0, U_{1}=1, U_{n}=PU_{n-1}-QU_{n-2} (n≥2). We show that the only sequence with U_{12}(P,Q) a perfect square is the Fibonacci sequence {U_{n}(1,-1)}; and we show that there are no non-degenerate sequences {U_{n}(P,Q)} with U_{9}(P,Q) a perfect square. The argument involves finding all rational points on several curves of genus 2.

Original language | English (US) |
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Pages (from-to) | 215-227 |

Number of pages | 13 |

Journal | Journal of Number Theory |

Volume | 107 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 2004 |

### Keywords

- Elliptic curve
- Formal group
- Lucas sequence

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Bremner, A., & Tzanakis, N. (2004). Lucas sequences whose 12th or 9th term is a square.

*Journal of Number Theory*,*107*(2), 215-227. https://doi.org/10.1016/j.jnt.2004.04.002