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Two-parameter families of uniquely extendable Diophantine triples

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Abstract

Let A and K be positive integers and є ∈ {−2;−1, 1, 2}. The main contribution of the paper is a proof that each of the D(є2)-triples {K,A2K + 2єA, (A + 1)2K + 2є(A + 1)} has unique extension to a D(є2)- quadruple. This is used to slightly strengthen the conditions required for the existence of a D(1)-quintuple whose smallest three elements form a regular triple.

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Acknowledgements

This work was supported by Grants-in-Aid for Scientific Research (JSPS KAKENHI) (Grant No. 16K05079).

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Authors and Affiliations

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

    Mihai Cipu

  2. Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, 276-0028, Japan

    Yasutsugu Fujita

  3. Département de Mathématique, Université de Strasbourg, Strasbourg, 67084, France

    Maurice Mignotte

Authors
  1. Mihai Cipu
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  2. Yasutsugu Fujita
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  3. Maurice Mignotte
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Correspondence to Yasutsugu Fujita.

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Cipu, M., Fujita, Y. & Mignotte, M. Two-parameter families of uniquely extendable Diophantine triples. Sci. China Math. 61, 421–438 (2018). https://doi.org/10.1007/s11425-015-0638-0

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  • DOI: https://doi.org/10.1007/s11425-015-0638-0

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