## Abstract

We pose a simple Diophantine problem which may be expressed in the language of geometry. Let C be a hyperelliptic curve given by the equation y^{2} = f(x), where f ∈ ℤ[x] is without multiple roots. We say that points P_{i} = (x_{i}; y_{i}) ∈ C(ℚ) for i = 1, 2, . ., k, are in geometric progression if the numbers x_{i} for i = 1, 2, . . ., k, are in geometric progression. Let n ≥ 3 be a given integer. In this paper we show that there exist polynomials a, b 2 ℤ[t] such that on the curve y^{2} = a(t)x^{n} + b(t) (defined over the field ℚ(t)) we can find four points in geometric progression. In particular this result generalizes earlier results of Berczes and Ziegler concerning the existence of geometric progressions on Pell type quadrics y^{2} = ax^{2} + b. We also investigate for fixed b ∈ ℤ, when there can exist rationals y_{i}, i = 1, . . ., 4, with -y^{2} _{i} - b} forming a geometric progression, with particular attention to the case b = 1. Finally, we show that there exist infinitely many parabolas y^{2} = ax + b which contain five points in geometric progression.

Original language | English (US) |
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Pages (from-to) | 669-683 |

Number of pages | 15 |

Journal | Publicationes Mathematicae |

Volume | 82 |

Issue number | 3-4 |

DOIs | |

State | Published - 2013 |

## Keywords

- Geometric progressions
- Hyperelliptic curves
- Rational points

## ASJC Scopus subject areas

- General Mathematics