There is no Diophantine quintuple
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- by Bo He, Alain Togbé and Volker Ziegler PDF
- Trans. Amer. Math. Soc. 371 (2019), 6665-6709 Request permission
Abstract:
A set of $m$ positive integers $\{a_1, a_2, \ldots , a_m\}$ is called a Diophantine $m$-tuple if $a_i a_j + 1$ is a perfect square for all $1 \le i < j \le m$. Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine $m$-tuples states that no Diophantine quintuple exists at all. We prove this conjecture.References
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Additional Information
- Bo He
- Affiliation: Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, People’s Republic of China; and Institute of Applied Mathematics, Aba Teachers University, Wenchuan, Sichuan 623000, People’s Republic of China
- MR Author ID: 825248
- Email: bhe@live.cn
- Alain Togbé
- Affiliation: Department of Mathematics, Statistics, and Computer Science, Purdue University Northwest, 1401 South U.S. Highway 421, Westville, Indiana
- Email: atogbe@pnw.edu
- Volker Ziegler
- Affiliation: Institute of mathematics, University of Salzburg, Hellbrunner Strasse 34/I, A-5020 Salzburg, Austria
- MR Author ID: 744740
- Email: volker.ziegler@sbg.ac.at
- Received by editor(s): October 27, 2017
- Received by editor(s) in revised form: March 1, 2018
- Published electronically: October 23, 2018
- Additional Notes: The first author was supported by Natural Science Foundation of China (Grant No. 11301363), and Sichuan provincial scientific research and innovation team in universities (Grant No. 14TD0040), and the Natural Science Foundation of Education Department of Sichuan Province (Grant No. 16ZA0371).
The second author thanks Purdue University Northwest for the partial support.
The third author was supported in part by the Austrian Science Fund (FWF) (Grant No. P 24801-N26). - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6665-6709
- MSC (2010): Primary 11B37, 11J86, 11Y50; Secondary 11D09, 11J68
- DOI: https://doi.org/10.1090/tran/7573
- MathSciNet review: 3937341