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Research articles

ScienceAsia 47 (2021): 106-110 |doi: 10.2306/scienceasia1513-1874.2021.005


On the divisibility Fk|Fx2 + Fx + 1


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