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The regularity of Diophantine quadruples


Authors: Yasutsugu Fujita and Takafumi Miyazaki
Journal: Trans. Amer. Math. Soc. 370 (2018), 3803-3831
MSC (2010): Primary 11D45; Secondary 11D09, 11B37, 11J68, 11J86
DOI: https://doi.org/10.1090/tran/7069
Published electronically: December 27, 2017
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Abstract: A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by the unity is a perfect square. A conjecture on the regularity of Diophantine quadruples asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element of the triple. The problem is reduced to studying an equation expressed as the coincidence of two linear recurrence sequences with initial terms composed of the fundamental solutions of some Pellian equations. In this paper, we determine the values of those initial terms completely and obtain finiteness results on the number of solutions of the equation. As one of the applications to the problem on the regularity of Diophantine quadruples, we show in general that the number of ways of extending any given Diophantine triple is at most $ 11$.


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Additional Information

Yasutsugu Fujita
Affiliation: Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
Email: fujita.yasutsugu@nihon-u.ac.jp

Takafumi Miyazaki
Affiliation: Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma, Japan
Email: tmiyazaki@gunma-u.ac.jp

DOI: https://doi.org/10.1090/tran/7069
Keywords: Diophantine tuples, system of Pellian equations, simultaneous rational approximation of irrationals, linear forms in logarithms
Received by editor(s): January 29, 2016
Received by editor(s) in revised form: August 23, 2016
Published electronically: December 27, 2017
Additional Notes: The first author was supported by JSPS KAKENHI Grant Number 16K05079
The second author was supported by JSPS KAKENHI Grant Number 16K17557
Article copyright: © Copyright 2017 American Mathematical Society

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