## The regularity of Diophantine quadruples

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- by Yasutsugu Fujita and Takafumi Miyazaki PDF
- Trans. Amer. Math. Soc.
**370**(2018), 3803-3831 Request permission

## 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 $11$.## References

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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
- MR Author ID: 720213
- ORCID: 0000-0001-7985-9667
- 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
- MR Author ID: 887630
- Email: tmiyazaki@gunma-u.ac.jp
- 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 - © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3811510