### Abstract

For a nonzero integer n, a set of distinct nonzero integers {a1,a2,…,am} such that aiaj+n is a perfect square for all 1≤i<j≤m, is called a Diophantine m-tuple with the property D(n) or simply D(n)-set. D(1)-sets are known as simply Diophantine m-tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine m-tuple (D(1)-set) which is also a D(n)-set for some n≠1. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophantine triples which are also D(n)-sets for some n≠1. However, the conjecture does not hold, since, for example, {8,21,55} is a D(1) and D(4321)-triple, while {1,8,120} is a D(1) and D(721)-triple. We present several infinite families of Diophantine triples {a,b,c} which are also D(n)-sets for two distinct n's with n≠1, as well as some Diophantine triples which are also D(n)-sets for three distinct n's with n≠1. We further consider some related questions.

Originalsprache | englisch |
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Aufsatznummer | https://doi.org/10.1016/j.jnt.2017.08.024 |

Seiten (von - bis) | 330-341 |

Seitenumfang | 12 |

Fachzeitschrift | Journal of Number Theory |

Jahrgang | 184 |

Publikationsstatus | Veröffentlicht - Mär 2018 |

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## Dieses zitieren

Kreso, D. (2018). Triples which are D(n)-sets for several n's.

*Journal of Number Theory*,*184*, 330-341. [https://doi.org/10.1016/j.jnt.2017.08.024].