### Abstract

Original language | English |
---|---|

Article number | https://doi.org/10.1016/j.jnt.2017.08.024 |

Pages (from-to) | 330-341 |

Number of pages | 12 |

Journal | Journal of Number Theory |

Volume | 184 |

Publication status | Published - Mar 2018 |

### Fingerprint

### Keywords

- Diophantine equations
- Diophantine m-tupules

### Cite this

*Journal of Number Theory*,

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

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

Research output: Contribution to journal › Article › Research › peer-review

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

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TY - JOUR

T1 - Triples which are D(n)-sets for several n's

AU - Kreso, Dijana

PY - 2018/3

Y1 - 2018/3

N2 - 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.

AB - 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.

KW - Diophantine equations

KW - Diophantine m-tupules

M3 - Article

VL - 184

SP - 330

EP - 341

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

M1 - https://doi.org/10.1016/j.jnt.2017.08.024

ER -