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

**Download full paper: PDF, 189 Kb**

## Details

### 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

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

## Related papers

## Cite this paper

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