On the number of extensions of a Diophantine triple-Reference-Cited by-同舟云学术

On the number of extensions of a Diophantine triple

Published:2018-03-25 Issue:03 Volume:14 Page:899-917
ISSN:1793-0421
Container-title:International Journal of Number Theory
language:en
Short-container-title:Int. J. Number Theory

Author:

Cipu Mihai¹,Fujita Yasutsugu²,Miyazaki Takafumi³

Affiliation:

1. Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit nr. 5, P. O. Box 1-764, RO-014700 Bucharest, Romania

2. Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan

3. Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma, Japan

Abstract

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

Link

https://www.worldscientific.com/doi/pdf/10.1142/S1793042118500549

Reference14 articles.

1. THE EQUATIONS 3x2−2 = y2 AND 8x2−7 = z2

2. ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k3− 4k}

3. Further remarks on Diophantine quintuples

4. Bounds for Diophantine quintuples II

Cited by 12 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. Extensions of a Diophantine triple by adjoining smaller elements II;Periodica Mathematica Hungarica;2023-12-14

2. Extensions of a Diophantine Triple by Adjoining Smaller Elements;Mediterranean Journal of Mathematics;2022-07-11

3. The extension of the D(−k)-triple $$\{1,k,k+1\}$$ to a quadruple;Acta Mathematica Hungarica;2022-04

4. There are no Diophantine quadruples of Pell numbers;International Journal of Number Theory;2021-07-01

5. The extensibility of the Diophantine triple {2, b, c};Analele Universitatii "Ovidius" Constanta - Seria Matematica;2021-06-01