Glasnik Matematicki, Vol. 59, No. 2 (2024), 259-276. \( \)

DIOPHANTINE \(D(n)\)-QUADRUPLES IN \(\mathbb{Z}[\sqrt{4k + 2}]\)

Kalyan Chakraborty, Shubham Gupta and Azizul Hoque

Department of Mathematics, SRM University AP, Neerukonda, Mangalagiri, Guntur-522240, Andhra Pradesh, India
e-mail:kalychak@gmail.com & kalyan.c@srmap.edu.in

Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj - 211019, India
e-mail:shubhamgupta2587@gmail.com

Department of Mathematics, Faculty of Science, Rangapara College, Rangapara, Sonitpur-784505, Assam, India
e-mail:ahoque.ms@gmail.com

Abstract.   Let \(d\) be a square-free integer and \(\mathbb{Z}[\sqrt{d}]\) a quadratic ring of integers. For a given \(n\in\mathbb{Z}[\sqrt{d}]\), a set of \(m\) non-zero distinct elements in \(\mathbb{Z}[\sqrt{d}]\) is called a Diophantine \(D(n)\)-\(m\)-tuple (or simply \(D(n)\)-\(m\)-tuple) in \(\mathbb{Z}[\sqrt{d}]\) if product of any two of them plus \(n\) is a square in \(\mathbb{Z}[\sqrt{d}]\). Assume that \(d \equiv 2 \pmod 4\) is a positive integer such that \(x^2 - dy^2 = -1\) and \(x^2 - dy^2 = 6\) are solvable in integers. In this paper, we prove the existence of infinitely many \(D(n)\)-quadruples in \(\mathbb{Z}[\sqrt{d}]\) for \(n = 4m + 4k\sqrt{d}\) with \(m, k \in \mathbb{Z}\) satisfying \(m \not\equiv 5 \pmod{6}\) and \(k \not\equiv 3 \pmod{6}\). Moreover, we prove the same for \(n = (4m + 2) + 4k\sqrt{d}\) when either \(m \not\equiv 9 \pmod{12}\) and \(k \not\equiv 3 \pmod{6}\), or \(m \not\equiv 0 \pmod{12}\) and \(k \not\equiv 0 \pmod{6}\). At the end, some examples supporting the existence of quadruples in \(\mathbb{Z}[\sqrt{d}]\) with the property \(D(n)\) for the above exceptional \(n\)'s are provided for \(d = 10\).

2020 Mathematics Subject Classification.   11D09, 11R11

Key words and phrases.   Diophantine quadruples, Pellian equations, quadratic fields

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https://doi.org/10.3336/gm.59.2.01

References:

  1. F. S. Abu Muriefah and A. Al Rashed, Some Diophantine quadruples in the ring \(\mathbb{Z}[\sqrt{-2}]\), Math. Commun. 9 (2004), 1–8.
    MathSciNet    Link

  2. N. Adžaga, On the size of Diophantine \(m\)-tuples in imaginary quadratic number rings, Bull. Math. Sci. 11 (2021), Article ID: 1950020, 10pp.
    MathSciNet    CrossRef

  3. A. Baker and H. Davenport, The equations \(3x^2 - 2 = y^2\) and \(8x^2 - 7 = z^2\), Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137.
    MathSciNet    CrossRef

  4. M. Bliznac Trebješanin and A. Filipin, Nonexistence of \(D(4)\)-quintuples, J. Number Theory 194 (2019), 170–217.
    MathSciNet    CrossRef

  5. N. C. Bonciocat, M. Cipu and M. Mignotte, There is no Diophantine \(D(-1)\)-quadruple, J. London Math. Soc. (2) 105 (2022), 63–99.
    MathSciNet    CrossRef

  6. E. Brown, Sets in which \(xy + k\) is always a square, Math. Comp. 45 (1985), 613–620.
    MathSciNet    CrossRef

  7. K. Chakraborty, S. Gupta, and A. Hoque, On a conjecture of Franušić and Jadrijević: counter-examples, Results Math. 78 (2023), article no. 18, 14pp.
    MathSciNet    CrossRef

  8. K. Chakraborty, S. Gupta and A. Hoque, Diophantine triples with the property \(D(n)\) for distinct \(n\)'s, Mediterr. J. Math. 20 (2023), article no. 31, 13pp.
    MathSciNet    CrossRef

  9. A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15–27.
    MathSciNet    CrossRef

  10. A. Dujella, Some polynomial formulas for Diophantine quadruples, Grazer Math. Ber. 328 (1996), 25–30.
    MathSciNet

  11. A. Dujella, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32 (1997), 1–10.
    MathSciNet   

  12. A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183–214.
    MathSciNet    CrossRef

  13. A. Dujella, Number theory, Školska knjiga, Zagreb, 2021.
    MathSciNet    Link

  14. A. Dujella, Diophantine \(m\)-tuples and elliptic curves, Springer, Cham, 2024.
    MathSciNet    CrossRef

  15. A. Dujella, Triples, quadruples and quintuples which are \(D(n)\)-sets for several \(n\)'s, in: Class groups of number fields and related topics (K. Chakraborty, A. Hoque and P. P. Pandey, Eds.), Springer, Cham, 2024, 77–92.
    CrossRef

  16. C. Elsholz, A. Filipin and Y. Fujita, On Diophantine quintuples and \(D(-1)\)-quadruples, Monats. Math. 175 (2014), 227–239.
    MathSciNet    CrossRef

  17. Z. Franušić, Diophantine quadruples in the ring \(\mathbb{Z}[\sqrt{2}]\), Math. Commun. 9 (2004), 141–148.
    MathSciNet    Link

  18. Z. Franušić, Diophantine quadruples in \(\mathbb{Z}[\sqrt{4k + 3}]\), Ramanujan J. 17 (2008), 77–88.
    MathSciNet    CrossRef

  19. Z. Franušić, A Diophantine problem in \(\mathbb{Z}[\sqrt{(1 + d)/2}]\), Studia Sci. Math. Hungar. 46 (2009), 103–112.
    MathSciNet    CrossRef

  20. Z. Franušić, Diophantine quadruples in the ring of integers of the pure cubic field \(\mathbb{Q}(\sqrt[3]{2})\), Miskolc Math. Notes 14 (2013), 893–903.
    MathSciNet    CrossRef

  21. Z. Franušić and I. Soldo, The problem of Diophantus for integers of \(\mathbb{Q}(\sqrt{-3})\), Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18 (2014), 15–25.
    MathSciNet    Link

  22. Z. Franušić and B. Jadrijević, \(D(n)\)-quadruples in the ring of integers of \(\mathbb{Q}(\sqrt{2}, \sqrt{3})\), Math. Slovaca 69 (2019), 1263–1278.
    MathSciNet    CrossRef

  23. S. Gupta, \(D(-1)\) tuples in imaginary quadratic fields, Acta Math. Hungar. 164 (2021), 556–569.
    MathSciNet    CrossRef

  24. B. He, A. Togbé and V. Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc. 371 (2019), 6665–6709.
    MathSciNet    CrossRef

  25. Lj. Jukić Matić, Non-existence of certain Diophantine quadruples in rings of integers of pure cubic fields, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), 163–167.
    MathSciNet    CrossRef

  26. I. Soldo, On the existence of Diophantine quadruples in \(\mathbb{Z}[\sqrt{-2}]\), Miskolc Math. Notes 14 (2013), 265–277.
    MathSciNet    CrossRef
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