On Diophantine triples and quadruples

Yifan Zhang and G. Grossman
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 4, Pages 6–16
Full paper (PDF, 196 Kb)

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Authors and affiliations

Yifan Zhang
Department of Mathematics, Central Michigan University
Mount Pleasant, MI, USA

G. Grossman
Department of of Mathematics, Central Michigan University
Mount Pleasant, MI, USA

Abstract

In this paper we consider Diophantine triples {a, b, c} (denoted D(n)-3-tuples) and give necessary and sufficient conditions for existence of integer n given the 3-tuple {a, b, c} so that ab + n, ac + n, bc + n are all squares of integers. Several examples as applications of the main results, related to both Diophantine triples and quadruples, are given.

Keywords

• Diophantine triples and quadruples
• Necessary and sufficient conditions

AMS Classification

• 11D99

References

1. Brown, E. (1985) Sets in which xy + k is always a square., Math. Comp. 45, 613–620.
2. Dujella, A. (1993) Generalization of a problem of Diophantus, Acta Arith. 65, 15–27.
3. Dujella, A. (1998) Complete solution of a family of simultaneous Pellian equations, Acta Math. Inform. Univ. Ostraviensis, 6, 59–67.
4. Dujella, A., A. Filipin, & C. Fuchs (2007) Effective solution of the D(−1)-quadruple conjecture, Acta Math. Inform. Univ. Ostraviensis,128(4), 319–338.
5. Dujella, A., & C. Fuchs (2005) Complete solution of a problem of Diophantus and Euler, Journal of the London Mathematical Society, 71(1), 33–52.
6. Filipin, A. (2005) Non-extendibility of D(− 1)-triples of the form {1, 10, c}, International Journal of Mathematics and Mathematical Sciences, 2005(4), 2217–2226.
7. Gibbs, P. (1999) A generalised stern-brocot tree from regular Diophantine quadruples, Arxiv preprint math/9903035.
8. Gupta, H., & K. Singh, On k-triad sequences, International Journal of Mathematics and Mathematical Sciences, 8(4), 799–804.
9. Kihel, O. (2000) On the extendibility of the P_{− 1}-set {1, 2, 5}, Fibonacci Quarterly, 38 (5), 464–466.
10. Mohanty, S. P, & A. M. S. Ramasamy (1984) The simultaneous Diophantine equations 5y² − 20 = x² and 2y² + 1 = z², Journal of Number Theory, 18(3), 356–359.
11. Mohanty, S. P., & A. M. S. Ramasamy (1985) On P_r,k sequences, Fibonacci Quarterly, 23(1), 36–44.
12. Abu Muriefah, F. S., & A. Al-Rashed, (2004) On the extendibility of the Diophantine triple {1, 5, c}, International Journal of Mathematics and Mathematical Sciences, 2004 (33), 1737–1746.
13. Walsh, P. G. (1999) On two classes of simultaneous Pell equations with no solutions, Mathematics of Computation, 68(225), 385–388.
14. Zhang, Y., & G. Grossman, presented Sixteenth International Conference on Fibonacci Numbers and their Applications, July 20-26, 2014, Rochester, New York, pre-print.
15. Zhang, Y. (2011) Masters thesis, Central Michigan University.