### 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) |
---|---|

Pages (from-to) | 511-520 |

Number of pages | 10 |

Journal | Journal of Number Theory |

Volume | 124 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2007 |

### Fingerprint

### Keywords

- Genus two curves
- Lucas sequence
- Squares

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

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

**On squares in Lucas sequences.** / Bremner, Andrew; Tzanakis, N.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 124, no. 2, pp. 511-520. https://doi.org/10.1016/j.jnt.2006.10.007

}

TY - JOUR

T1 - On squares in Lucas sequences

AU - Bremner, Andrew

AU - Tzanakis, N.

PY - 2007/6

Y1 - 2007/6

N2 - Let P and Q be non-zero integers. The Lucas sequence {Un (P, Q)} is defined by U0 = 0, U1 = 1, Un = P Un - 1 - Q Un - 2 (n ≥ 2). The question of when Un (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un 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 Un (P, Q) = □, are given by U8 (1, - 4) = 212, U8 (4, - 17) = 6202, and U12 (1, - 1) = 122.

AB - Let P and Q be non-zero integers. The Lucas sequence {Un (P, Q)} is defined by U0 = 0, U1 = 1, Un = P Un - 1 - Q Un - 2 (n ≥ 2). The question of when Un (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un 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 Un (P, Q) = □, are given by U8 (1, - 4) = 212, U8 (4, - 17) = 6202, and U12 (1, - 1) = 122.

KW - Genus two curves

KW - Lucas sequence

KW - Squares

UR - http://www.scopus.com/inward/record.url?scp=34047244951&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34047244951&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2006.10.007

DO - 10.1016/j.jnt.2006.10.007

M3 - Article

AN - SCOPUS:34047244951

VL - 124

SP - 511

EP - 520

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -