The extension of some D(4)-pairs

Alan Filipin
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 2, Pages 126—135
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Authors and affiliations

Alan Filipin
Faculty of Civil Engineering, University of Zagreb
Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia

Abstract

In this paper we illustrate the use of the results from [1] proving that D(4)-triple {a,b,c} with a < b < a+57√ a  has a unique extension to a quadruple with a larger element. This furthermore implies that D(4)-pair {a,b} cannot be extended to a a quintuple if a < b < a+57√ a .

Keywords

  • Diophantine m-tuples
  • Simultaneous Diophantine equations

AMS Classification

  • Primary 11D09
  • Secondary 11J68

References

  1. Baćić, Lj. & Filipin, A. (2013) The extendibility of D(4)-pairs, Math. Commun., 18, 447–456.
  2. Baćić, Lj. & Filipin, A. (2013) On the extendibility of D(4)-pair {k − 2, k + 2}, J. Comb. Number Theory, 5, 181–197.
  3. Baćić, Lj. & Filipin, A. (2014) A note on the number of D(4)-quintuples, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., 18, 7–13.
  4. Dujella, A. Diophantinem-tuples, http://web.math.pmf.unizg.hr/˜duje/dtuples.html.
  5. Dujella, A. & Ramasamy, A. M. S. (2005) Fibonacci numbers and sets with the property D(4), Bull. Belg. Math. Soc. Simon Stevin, 12, 401–412.
  6. Filipin, A. (2009) On the size of sets in which xy + 4 is always a square, Rocky Mountain J. Math., 39, 1195–1224.
  7. Filipin, A. (2008) There does not exist a D(4)-sextuple, J. Number Theory, 128, 1555–1565.
  8. Filipin, A. (2009) An irregular D(4)-quadruple cannot be extended to a quintuple, Acta Arith., 136, 167–176.
  9. Filipin, A., Fujita, Y. & Togbe, A. (2014) The extendibility of Diophantine pairs II: examples, J. Number Theory, 145, 604–631.
  10. Filipin, A., He, B. & Togbe, A. (2012) On a family of two-parametric D(4)-triples, Glas. Mat. Ser. III, 47, 31–51.
  11. He, B. & Togbe, A. (2011) On the D(−1)-triple {1, k2 + 1, k2 + 2k + 2} and its unique D(1)-extension, J. Number Theory, 131, 120–137.
  12. Mignotte, M. (1998) A corollary to a theorem of Laurent–Mignotte–Nesterenko, Acta Arith., 86, 101–111.

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Cite this paper

APA

Filipin, A. (2017). The extension of some D(4)-pairs. Notes on Number Theory and Discrete Mathematics, 23(2), 126-135.

Chicago

Filipin, Alan. “The Extension of Some D(4)-pairs.” Notes on Number Theory and Discrete Mathematics 23, no. 2 (2017): 126-135.

MLA

Filipin, Alan.. “The Extension of Some D(4)-pairs.” Notes on Number Theory and Discrete Mathematics 23.2 (2017): 126-135. Print.

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