On the size of Diophantine m-tuples in imaginary quadratic number rings

Scientific paper - Original scientific paper

• Author: Nikola Adžaga
• Author's institution: University of Zagreb; Faculty of Civil Engineering
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Civil Engineering

Abstract (english)

A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that m≤42. Our proof is relatively simple compared to the proofs of similar results in positive integers.

• Keywords (english): Diophantine m-tuples; Diophantine approximation; Pell equations; rings of integers; Diophantine equations
• Language: english
• Publication type: Scientific paper - Original scientific paper
• Publication status: Published
• Peer review: Peer review - international
• Publication version: Published version
• Journal title: Bulletin of Mathematical Sciences
• Numbering: vol. 11(1), 1950020
• e-ISSN: 1664-3615
• DOI: https://doi.org/10.1142/S1664360719500206
• URN:NBN: urn:nbn:hr:237:032005
• Publication: 2021
• Type of resource: Text
• Access conditions: Open access
• Created on: 2024-09-04 10:57:15