Abstract
Let K be an imaginary quadratic number field, and \( \mathcal{O}_K\) its ring of integers. In this article, we prove the non-existence of Diophantine \(m\)-tuples in \(\mathcal{O}_K\) with the property \(D(-1)\), for \(m > 36\).
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References
N. Adžaga, On the size of Diophantine \(m\)-tuples in imaginary quadratic number rings, Bull. Math. Sci., 9 (2019) 1950020, 10 pp
Bonciocat, N.C., Cipu, M., Mignotte, M.: On \(D(-1)\)-quadruples. Publ. Mat. 56, 279–304 (2012)
N. C. Bonciocat, M. Cipu and M. Mignotte, There is no Diophantine \(D(-1)\)-quadruple, arXiv:2010.09200
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
Bennett, M.A.: On the number of solutions of simultaneous Pell equations. J. Reine Angew. Math. 498, 173–199 (1998)
Dujella, A., Fuchs, C.: Complete solution of a problem of Diophantus and Euler. J. London Math. Soc. 71, 33–52 (2005)
Dujella, A.: The problem of Diophantus and Davenport for Gaussian integers. Glas. Mat. Ser. III(32), 1–10 (1997)
Dujella, A.: There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566, 183–214 (2004)
Dujella, A., Filipin, A., Fuchs, C.: Effective solution of the \(D(-1)\)-quadruple conjecture. Acta Arith. 128, 319–338 (2007)
Elsholtz, C., Filipin, A., Fujita, Y.: On Diophantine quintuples and \(D(-1)\)-quadruples. Monatsh. Math. 175(2), 227–239 (2014)
Filipin, A., Fujita, Y.: The number of \(D(-1)\)-quadruples. Math. Commun. 15, 387–391 (2010)
He, B., Togbé, A., Ziegler, V.: There is no Diophantine quintuple. Trans. Amer. Math. Soc. 371, 6665–6709 (2019)
Jadrijević, B., Ziegler, V.: A system of relative Pellian equations and a related family of relative Thue equations. Int. J. Number Theory 2, 569–590 (2006)
Lapkova, K.: Explicit upper bound for an average number of divisors of quadratic polynomials. Arch. Math. (Basel) 106, 247–256 (2016)
K. Lapkova, Correction to: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials, Monatsh. Math., 186 (4), 675-678
Trudgian, T.: Bounds on the number of Diophantine quintuples. J. Number Theory 157, 233–249 (2015)
Acknowledgements
The author is indebted to Professor Kalyan Chakraborty for suggestions and for carefully going through the manuscript. The author is also thankful to Dr. Azizul Hoque for introducing him to this area and for his encouragement throughout. The author is grateful to the anonymous referee for valuable suggestions and remarks which improved the exposition of the paper.
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Gupta, S. \(D(-1)\) tuples in imaginary quadratic fields. Acta Math. Hungar. 164, 556–569 (2021). https://doi.org/10.1007/s10474-021-01150-w
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DOI: https://doi.org/10.1007/s10474-021-01150-w