### Abstract

Let P and Q be non-zero integers. The Lucas sequence {U_{n} (P, Q)} is defined by U_{0} = 0, U_{1} = 1, U_{n} = P U_{n - 1} - Q U_{n - 2} (n ≥ 2). The question of when U_{n} (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, U_{n} is a square for infinitely many pairs (P, Q) with gcd (P, Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd (P, Q) = 1 and U_{n} (P, Q) = □, are given by U_{8} (1, - 4) = 21^{2}, U_{8} (4, - 17) = 620^{2}, and U_{12} (1, - 1) = 12^{2}.

Original language | English (US) |
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Pages (from-to) | 511-520 |

Number of pages | 10 |

Journal | Journal of Number Theory |

Volume | 124 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2007 |

### Keywords

- Genus two curves
- Lucas sequence
- Squares

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Bremner, A., & Tzanakis, N. (2007). On squares in Lucas sequences.

*Journal of Number Theory*,*124*(2), 511-520. https://doi.org/10.1016/j.jnt.2006.10.007