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 \(5\cdot 10^{60}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arkin, J., Hoggatt, V.E., Strauss, E.G.: On Euler’s solution of a problem of Diophantus. Fibonacci Quart. 17, 333–339 (1979)
Bonciocat, N.C., Cipu, M., Mignotte, M.: On \(D(-1)\)-quadruples. Publ. Mat. 56, 279–304 (2012)
Bordellès, O.: An inequality for the class number. J. Ineq. Pure Appl. Math. 7(87), 1–8 (2006)
Dujella, A.: An absolute bound for the size of Diophantine \(m\)-tuples. J. Number Theory 89, 126–150 (2001)
Dujella, A.: There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566, 183–214 (2004)
Dujella, A.: On the number of Diophantine \(m\)-tuples. Ramanujan J. 15, 37–46 (2008)
Dujella, A.: webpage. http://web.math.pmf.unizg.hr/~duje/dtuples.html
Dujella, A., Filipin, A., Fuchs, C.: Effective solution of the \(D(-1)\)-quadruple conjecture. Acta Arith. 128, 319–338 (2007)
Dujella, A., Fuchs, C.: Complete solution of a problem of Diophantus and Euler. J. Lond. Math. Soc. 71, 33–52 (2005)
Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. (2) 49, 291–306 (1998)
Filipin, A., Fujita, Y.: The number of \(D(-1)\)-quadruples. Math. Commun. 15, 387–391 (2010)
Filipin, A., Fujita, Y.: The number of Diophantine quintuples II. Publ. Math. Debrecen. 82, 293–308 (2013)
Fujita, Y.: Any Diophantine quintuple contains a regular Diophantine quadruple. J. Number Theory 129, 1678–1697 (2009)
Fujita, Y.: The number of Diophantine quintuples. Glas. Mat. Ser. III 45, 15–29 (2010)
Hooley, C.: On the number of divisors of quadratic polynomials. Acta Math. 110, 97–114 (1963)
Jones, B.W.: A second variation on a problem of Diophantus and Davenport. Fibonacci Quart. 16, 155–165 (1978)
Martin, G., Sitar, S.: Erdős-Turán with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples. Mathematika 57, 1–29 (2011)
McKee, J.: On the average number of divisors of quadratic polynomials. Math. Proc. Camb. Philos. Soc. 117, 389–392 (1995)
Vinogradov, I.M.: Elements of number theory. Dover, New York (1954)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Schoißengeier.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-013-0571-5