Rad HAZU, Matematičke znanosti, Vol. 18 (2014), 7-13.

A NOTE ON THE NUMBER OF D(4)-QUINTUPLES

Ljubica Baćić and Alan Filipin

Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia
e-mail: filipin@master.grad.hr

Abstract.   In this paper we will significantly improve the known bound on the number of D(4)-quintuples, illustrating the elegant use of the results the authors proved in [1] together with more efficient way of counting the number of m-tuples that was introduced in [5]. More precisely, we prove that there are at most 7 · 10³⁶ D(4)-quintuples.

2010 Mathematics Subject Classification.   11D09, 11J68.

Key words and phrases.   Diophantine tuples, simultaneous Diophantine equations.

References:

1. Lj. Baćić and A. Filipin, On the extendibility of D(4)-pairs, Math. Communn. 18 (2013), 447-456.
   MathSciNet

2. A. Dujella, Diophantine m-tuples, http://web.math.pmf.unizg.hr/~duje/dtuples.html.

3. A. Dujella and M. Mikić, On the torsion group of elliptic curves induced by D(4)-triples, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 22 (2014), 79-90.
   MathSciNet

4. A. Dujella and A. M. S. Ramasamy, Fibonacci numbers and sets with the property D(4), Bull. Belg. Math. Soc. Simon Stevin 12(3) (2005), 401-412.
   MathSciNet

5. C. Elsholtz, A. Filipin and Y. Fujita, On Diophantine quintuples and D(-1)-quadruples, Monatsh. Math. 175 (2014), 227-239.
   MathSciNet     CrossRef

6. A. Filipin, There does not exist a D(4)-sextuple, J. Number Theory 128 (2008), 1555-1565.
   MathSciNet     CrossRef

7. A. Filipin, An irregular D(4)-quadruple cannot be extended to a quintuple, Acta Arith. 136 (2009), 167-176.
   MathSciNet     CrossRef

8. A. Filipin, There are only finitely many D(4)-quintuples, Rocky Mountain J. Math. 41 (2011), 1847-1860.
   MathSciNet     CrossRef

9. I. M. Vinogradov, Elements of number theory, Dover, New York, 1954.
   MathSciNet

10. W. Wu and Bo He, On Diophantine quintuple conjecture, Proc. Japan Acad. A Math. Sci. 90 (2014), 84-86.
    MathSciNet     CrossRef