On the divisibility F_k|F_x² + F_x + 1

Florian Luca^a,d,e, Prapanpong Pongsriiam^b,*, László Szalay^c,f

ABSTRACT: Let F n and Ln be the nth Fibonacci and Lucas numbers, respectively. We show that if Fk j Fx2 + Fx +1, then k 2 f4,7g; if Lk j Fx2 + Fx + 1, then k 2 f2, 4g.

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^a	School of Maths, Wits University, Johannesburg, South Africa
^b	Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000 Thailand
^c	Department of Mathematics, J. Selye University, Komarno 94501 Slovakia
^d	Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah 21589 Saudi Arabia
^e	Centro de Ciencias Matem?ticas, UNAM, Morelia 58089 Mexico
^f	Institute of Mathematics, University of Sopron, Sopron 9400 Hungary

* Corresponding author, E-mail: pongsriiam_p@silpakorn.edu

Received 5 Jun 2020, Accepted 24 Nov 2020

On the divisibility Fk|Fx2 + Fx + 1

Florian Lucaa,d,e, Prapanpong Pongsriiamb,*, László Szalayc,f

On the divisibility F_k|F_x² + F_x + 1

Florian Luca^a,d,e, Prapanpong Pongsriiam^b,*, László Szalay^c,f