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

Dijana Kreso

Research output: Contribution to journalArticleResearchpeer-review

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.
Original languageEnglish
Article numberhttps://doi.org/10.1016/j.jnt.2017.08.024
Pages (from-to)330-341
Number of pages12
JournalJournal of Number Theory
Volume184
Publication statusPublished - Mar 2018

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Keywords

  • Diophantine equations
  • Diophantine m-tupules

Cite this

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].

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

In: Journal of Number Theory, Vol. 184, https://doi.org/10.1016/j.jnt.2017.08.024, 03.2018, p. 330-341.

Research output: Contribution to journalArticleResearchpeer-review

Kreso, D 2018, 'Triples which are D(n)-sets for several n's' Journal of Number Theory, vol. 184, https://doi.org/10.1016/j.jnt.2017.08.024, pp. 330-341.
Kreso D. Triples which are D(n)-sets for several n's. Journal of Number Theory. 2018 Mar;184:330-341. https://doi.org/10.1016/j.jnt.2017.08.024.
Kreso, Dijana. / Triples which are D(n)-sets for several n's. In: Journal of Number Theory. 2018 ; Vol. 184. pp. 330-341.
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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.

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