### Abstract

We consider the family of relative Thue equations x^{3} - (t - 1)x^{2}y - (t + 2)xy^{2} - y^{3} = μ, where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. We prove that there are only trivial solutions (with x , y ≤ 1), if t is large enough or if the discriminant of the quadratic number field is large enough or if Ret = -1/2 (there are a few more solutions in this case which are explicitly listed). In the case Ret = -1/2, an algebraic method is used, in the general case, Baker's method yields the result.

Original language | English |
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Pages (from-to) | 437-449 |

Number of pages | 13 |

Journal | Journal of Symbolic Computation |

Volume | 34 |

Issue number | 5 |

DOIs | |

Publication status | Published - Nov 1 2002 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics

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## Cite this

Heuberger, C., Petho, A., & Tichy, R. F. (2002). Thomas' family of thue equations over imaginary quadratic fields.

*Journal of Symbolic Computation*,*34*(5), 437-449. https://doi.org/10.1006/jsco.2002.0568