Glasnik Matematicki, Vol. 52, No. 1 (2017), 11-21.

A BROCARD-RAMANUJAN-TYPE EQUATION WITH LUCAS AND ASSOCIATED LUCAS SEQUENCES

István Pink and Márton Szikszai

Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary and University of Salzburg, Hellbrunnerstrasse 34/I, A-5020 Salzburg, Austria
e-mail: pinki@science.unideb.hu & istvan.pink@sbg.ac.at

Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
e-mail: szikszai.marton@science.unideb.hu

Abstract.   This paper deals with a Brocard-Ramanujan-type equation of the form

un1un2 ⋯ unk+1=um2
in unknown nonnegative integers k,n1,n2, …,nk and m with k≥ 1, where u=(un)n=0∞ is either a Lucas sequence or its associated sequence. For certain infinite families of sequences we completely solve the above equation, extending some results of Marques [15], Szalay [21] and Pongsriiam [18]. The ingredients of the proofs are factorization properties of Lucas sequences, the celebrated result of Bilu, Hanrot and Voutier on primitive divisors of Lucas sequences and elementary estimations concerning the terms involved.

2010 Mathematics Subject Classification.   11B37, 11B39.

Key words and phrases.   Brocard-Ramanujan equation, Lucas sequences.

Full text (PDF) (free access)

DOI: 10.3336/gm.52.1.02

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