Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.6

On Arithmetic Progressions in Lucas Sequences


Lajos Hajdu
Institute of Mathematics
University of Debrecen
P. O. Box 400
H-4002 Debrecen
Hungary

Márton Szikszai
Institute of Mathematics, University of Debrecen
and
MTA-DE Research Group Equations Functions and Curves
Hungarian Academy of Sciences and University of Debrecen
P. O. Box 400
H-4002 Debrecen
Hungary

Volker Ziegler
Institute of Mathematics
University of Salzburg
Hellbrunnerstrasse 34/I
A-5020 Salzburg
Austria

Abstract:

In this paper, we consider arithmetic progressions contained in Lucas sequences of the first and second kind. We prove that for almost all Lucas sequences, there are only finitely many arithmetic three term progressions and their number can be effectively bounded. We also show that there are only a few Lucas sequences which contain infinitely many arithmetic three term progressions and one can explicitly list both the sequences and the progressions in them. A more precise statement is given for sequences with dominant zero.


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(Concerned with sequences A000032 A000045 A001045 A014551 A039834.)


Received March 20 2017; revised versions received August 7 2017; August 11 2017. Published in Journal of Integer Sequences, September 3 2017.


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