Abstract
Let \(d\equiv 2\pmod 4\) be a square-free integer such that \(x^2 - dy^2 =- 1\) and \(x^2 - dy^2 = 6\) are solvable in integers. We prove the existence of infinitely many quadruples in \({\mathbb {Z}}[\sqrt{d}]\) with the property D(n) when \(n \in \{(4m + 1) + 4k\sqrt{d}, (4m + 1) + (4k + 2)\sqrt{d}, (4m + 3) + 4k\sqrt{d}, (4m + 3) + (4k + 2)\sqrt{d}, (4m + 2) + (4k + 2)\sqrt{d}\}\) for \(m, k \in {\mathbb {Z}}\). As a consequence, we provide few counter examples to Conjecture 1 of Franušić and Jadrijević [Math. Slovaca 69, 1263–1278 (2019)].
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References
Abu Muriefah, F.S., Al Rashed, A.: Some diophantine quadruples in the ring \({\mathbb{Z} }[\sqrt{-2}]\). Math. Commun. 9, 1–8 (2004)
Baker, A., Davenport, H.: The equations \(3x^2 - 2 = y^2\) and \(8x^2 - 7 = z^2\). Q. J. Math. Oxf. Ser. (2) 20, 129–137 (1969)
Bonciocat, N.C., Cipu, M., Mignotte, M.: There is no diophantine \(D(-1)\)-quadruple. J. Lond. Math. Soc. 105, 63–69 (2022)
Brown, E.: Sets in which \(xy + k\) is always a square. Math. Comput. 45, 613–620 (1985)
Buniakovsky, V.: Sur les diviseurs numeriques invariables des fonctions rationnelles entieres. Mem. Acad. Sci. St. Petersb. 6, 305–329 (1857)
Chakraborty, K., Gupta, S., Hoque, A.: Diophantine triples with the property \(D(n)\) for distinct \(n\)’s. Mediterr. J. Math. (to appear)
Dujella, A.: Generalization of a problem of Diophantus. Acta Arith. 65, 15–27 (1993)
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.: Franušić, On differences of two squares in some quadratic fields. Rocky Mt. J. Math. 37, 429–453 (2007)
Dujella, A.: Number Theory. Školska knjiga, Zagreb (2021)
Elsholz, C., Filipin, A., Fujita, Y.: On diophantine quintuples and \(D(-1)\)-quadruples. Mon. Math. 175, 227–239 (2014)
Franušić, Z.: Diophantine quadruples in the ring \({\mathbb{Z} }[\sqrt{2}]\). Math. Commun. 9, 141–148 (2004)
Franušić, Z.: Diophantine quadruples in \({\mathbb{Z} }[\sqrt{4k + 3}]\). Ramanujan J. 17, 77–88 (2008)
Franušić, Z.: A Diophantine problem in \({\mathbb{Z} }[\sqrt{(1 + d)/2}]\). Studia Sci. Math. Hung. 46, 103–112 (2009)
Franušić, Z.: Diophantine quadruples in the ring of integers of the pure cubic field \({\mathbb{Q} }(\root 3 \of {2})\). Miskolc Math. Notes 14, 893–903 (2013)
Franušić, Z., Jadrijević, B.: \(D(n)\)-quadruples in the ring of integers of \({\mathbb{Q} }(\sqrt{2}, \sqrt{3})\). Math. Slovaca 69, 1263–1278 (2019)
Franušić, Z., Soldo, I.: The problem of Diophantus for integers of \({\mathbb{Q} }(\sqrt{-3})\). Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18, 15–25 (2014)
He, B., Togbé, A.: On the \(D(-1)\)-triple \(\{1, k^2+1, k^2+2k + 2\}\) and its unique \(D(1)\)-extension. J. Number Theory 131, 120–137 (2011)
He, B., Togbé, A., Ziegler, V.: There is no diophantine quintuple. Trans. Am. Math. Soc. 371, 6665–6709 (2019)
Lemmermeyer, F.: Higher descent on Pell conics I. From Legendre to Selmer, preprint. arXiv:math/0311309
Matić, L.J.: Non-existence of certain Diophantine quadruples in rings of integers of pure cubic fields. Proc. Jpn. Acad. Ser. A Math. Sci. 88(10), 163–167 (2012)
Mertens, F.: Üeber Dirichletsche Reihen. Ak. Wiss. Wien. (Math.) 104, 1093–1153 (1895)
Meyer, A.: Üeber einen Satz von Dirichlet. J. Reine Angew. Math. 103, 98–117 (1888)
Mohanty, S.P., Ramamsamy, M.S.: On \(P_{r, k}\) sequences. Fibonacci Q. 23, 36–44 (1985)
Soldo, I.: On the existence of Diophantine quadruples in \({\mathbb{Z} }[\sqrt{-2}]\). Miskolc Math. Notes 14, 265–277 (2013)
Acknowledgements
The authors thank the anonymous referee for his/her valuable comments/suggestions that immensely improved the results as well as the presentation of the paper. The third author would like to appreciate the hospitality provided by Kerala School of Mathematics, Kozhikode, Kerala, where the a part of the work was done. The third author acknowledges SERB MATRICS grant (No. MTR/2021/00762), Govt. of India.
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Chakraborty, K., Gupta, S. & Hoque, A. On a Conjecture of Franušić and Jadrijević: Counter-Examples. Results Math 78, 18 (2023). https://doi.org/10.1007/s00025-022-01794-2
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DOI: https://doi.org/10.1007/s00025-022-01794-2