Diophantine pairs that induce certain Diophantine triples

Scientific paper - Original scientific paper

• Author: Mihai Cipu
• Author: Alan Filipin
• Author: Yasutsugu Fujita
• Author's institution: University of Zagreb; Faculty of Civil Engineering
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract (english)

Diophantine tuples are sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. In the main theorem of this paper it is shown that any Diophantine triple, the second largest element of which is between the square and four times the square of the smallest one, is uniquely extended to a Diophantine quadruple by joining an element exceeding the largest element in the triple. A similar result is obtained under the hypothesis that the two smallest elements have the form T2+2T, 4T4+8T3−4Tfor some positive integer T, which we encounter as an exceptional case. The main theorem implies that the same is valid for triples with smallest elements KA2, 4KA4±4Afor some positive integers Aand K∈{1, 2, 3, 4}.

• Keywords (english): Diophantine m-tuples, Pellian equations, Linear forms in logarithms
• Language: english
• Publication type: Scientific paper - Original scientific paper
• Publication status: Published
• Peer review: Peer review - international
• Publication version: Published version
• Journal title: Journal of number theory
• Numbering: vol. 210, pp. 433-475
• e-ISSN: 1096-1658
• URN:NBN: urn:nbn:hr:237:853491
• Publication: 2020
• Type of resource: Text
• Access conditions: Open access
• Created on: 2024-09-05 06:16:37