Bliznac, Marija; Filipin, Alan An upper bound for the number of diophantine quintuples. (English) Zbl 1419.11052 Bull. Aust. Math. Soc. 94, No. 3, 384-394 (2016). Summary: We improve the known upper bound for the number of Diophantine \(D(4)\)-quintuples by using the most recent methods that were developed in the \(D(1)\) case. More precisely, we prove that there are at most \(6.8587\times 10^{29}\) \(D(4)\)-quintuples. Cited in 3 Documents MSC: 11D09 Quadratic and bilinear Diophantine equations 11D45 Counting solutions of Diophantine equations 11J86 Linear forms in logarithms; Baker’s method Keywords:diophantine \(m\)-tuples; Pell equations PDFBibTeX XMLCite \textit{M. Bliznac} and \textit{A. Filipin}, Bull. Aust. Math. Soc. 94, No. 3, 384--394 (2016; Zbl 1419.11052) Full Text: DOI References: [1] Cipu, Acta Arith. 173 pp 365– (2016) [2] DOI: 10.4064/aa168-3-1 · Zbl 1347.11028 · doi:10.4064/aa168-3-1 [3] Baćić, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18 pp 7– (2014) [4] Baćić, Math. Commun. 18 pp 447– (2013) [5] DOI: 10.1070/IM2008v072n06ABEH002429 · Zbl 1214.11088 · doi:10.1070/IM2008v072n06ABEH002429 [6] Vinogradov, Elements of Number Theory (1954) [7] DOI: 10.1016/j.jnt.2015.05.004 · Zbl 1367.11036 · doi:10.1016/j.jnt.2015.05.004 [8] DOI: 10.1216/RMJ-2011-41-6-1847 · Zbl 1237.11014 · doi:10.1216/RMJ-2011-41-6-1847 [9] DOI: 10.4064/aa136-2-5 · Zbl 1228.11036 · doi:10.4064/aa136-2-5 [10] DOI: 10.1216/RMJ-2009-39-4-1195 · Zbl 1230.11037 · doi:10.1216/RMJ-2009-39-4-1195 [11] Dujella, Bull. Belg. Math. Soc. Simon Stevin 12 pp 401– (2005) [12] Dujella, An. Ştiinţ. Univ. ”Ovidius” Constanţa Ser. Mat. 22 pp 79– (2014) [13] DOI: 10.1017/S0004972715001136 · Zbl 1336.11067 · doi:10.1017/S0004972715001136 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.