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On Diophantine quintuples and \(D(-1)\)-quadruples

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Abstract

In this paper the known upper bound \(10^{96}\) for the number of Diophantine quintuples is reduced to \(6.8\cdot 10^{32}\). The key ingredient for the improvement is that certain individual bounds on parameters are now combined with a more efficient counting of tuples, and estimated by sums over divisor functions. As a side effect, we also improve the known upper bound \(4\cdot 10^{70}\) for the number of \(D(-1)\)-quadruples to \(5\cdot 10^{60}\).

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Acknowledgments

The authors would like to thank Andrej Dujella for discussions on the subject, and the referee for a careful reading of the manuscript. The first author was partially supported by the Austrian Science Fund (FWF): W1230, the second author was supported by the Ministry of Science, Education and Sports, Republic of Croatia, grant 037-0372781-2821 and the third author was partially supported by JSPS KAKENHI Grant Number 25400025.

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Correspondence to Christian Elsholtz.

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Communicated by J. Schoißengeier.

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Elsholtz, C., Filipin, A. & Fujita, Y. On Diophantine quintuples and \(D(-1)\)-quadruples. Monatsh Math 175, 227–239 (2014). https://doi.org/10.1007/s00605-013-0571-5

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  • DOI: https://doi.org/10.1007/s00605-013-0571-5

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