Glasnik Matematicki, Vol. 49, No. 2 (2014), 303-312.

DIOPHANTINE TRIPLES AND REDUCED QUADRUPLES WITH THE LUCAS SEQUENCE OF RECURRENCE un=Aun-1-un-2

Nurettin Irmak and László Szalay

Mathematics Department, Art and Science Faculty, University of Niğde, 51240 Niğde, Turkey
e-mail: nirmak@nigde.edu.tr, irmaknurettin@gmail.com

Institute of Mathematics, University of West Hungary, H-9400 Sopron, Hungary
e-mail: laszlo.szalay@emk.nyme.hu


Abstract.   In this study, we show that there is no positive integer triple {a, b, c} such that all of ab+1, ac+1 and bc+1 are in the sequence {un}n≥ 0 satisfies the recurrence un=Aun-1-un-2 with the initial values u0=0, u1=1. Further, we investigate the analogous question for the quadruples {a,b,c,d} with abc+1=ux, bcd+1=uy, cda+1=uz and dab+1=ut, and deduce the non-existence of such quadruples.

2010 Mathematics Subject Classification.   11D72, 11B39.

Key words and phrases.   Diophantine triples, Diophantine quadruples, binary recurrence.


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DOI: 10.3336/gm.49.2.05


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