Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-21T00:50:14.400Z Has data issue: false hasContentIssue false
  • English
  • Français

AN UPPER BOUND FOR THE NUMBER OF DIOPHANTINE QUINTUPLES

Part of: Diophantine approximation, transcendental number theory Diophantine equations

Published online by Cambridge University Press:  16 August 2016

MARIJA BLIZNAC  and
ALAN FILIPIN
MARIJA BLIZNAC*
Affiliation:
Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia email marija.bliznac@grad.hr
ALAN FILIPIN
Affiliation:
Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia email filipin@grad.hr
*
✉marija.bliznac@grad.hr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Diophantine m-tuplesPell equations

MSC classification

Primary: 11D09: Quadratic and bilinear equations
Secondary: 11D45: Counting solutions of Diophantine equations 11J86: Linear forms in logarithms; Baker's method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 3 , December 2016 , pp. 384 - 394
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Aleksentsev, Y. M., ‘The Hilbert polynomial and linear forms in the logarithms of algebraic numbers’, Izv. Math. 72 (2008), 1063–1110.Google Scholar
Baćić, Lj. and Filipin, A., ‘The extensibility of D (4)-pairs’, Math. Commun. 18(2) (2013), 447–456.Google Scholar
Baćić, Lj. and Filipin, A., ‘A note on the number of D (4)-quintuples’, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18(519) (2014), 7–13.Google Scholar
Cipu, M., ‘Further remarks on Diophantine quintuples’, Acta Arith. 168(3) (2015), 201–219.Google Scholar
Cipu, M. and Trudgian, T., ‘Searching for Diophantine quintuples’, Acta Arith. 173 (2016), 365–382.Google Scholar
Dudek, A., ‘On the Number of Divisors of n 2 - 1’, Bull. Aust. Math. Soc. 93(2) (2016), 194–198.Google Scholar
Dujella, A., ‘Diophantine $m$ -tuples’, http://web.math.pmf.unizg.hr/∼duje/dtuples.html.Google Scholar
Dujella, A. and Mikić, M., ‘On the torsion group of elliptic curves induced by D (4)-triples’, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 22(2) (2014), 79–90.Google Scholar
Dujella, A. and Ramasamy, A. M. S., ‘Fibonacci numbers and sets with the property D (4)’, Bull. Belg. Math. Soc. Simon Stevin 12(3) (2005), 401–412.CrossRefGoogle Scholar
Filipin, A., ‘‘On the size of sets in which xy + 4 is always a square’’, Rocky Mountain J. Math. 39 (2009), 1195–1224.Google Scholar
Filipin, A., ‘An irregular D (4)-quadruple cannot be extended to a quintuple’, Acta Arith. 136(2) (2009), 167–176.CrossRefGoogle Scholar
Filipin, A., ‘There are only finitely many D (4)-quintuples’, Rocky Mountain J. Math. 41(6) (2011), 1847–1859.Google Scholar
Filipin, A., ‘The extension of some $D(4)$ -pairs’, submitted.Google Scholar
Trudgian, T., ‘Bounds on the number of Diophantine quintuples’, J. Number Theory 157 (2015), 233–249.CrossRefGoogle Scholar
Vinogradov, I. M., Elements of Number Theory (Dover, New York, 1954).Google Scholar