New York Journal of Mathematics
Volume 26 (2020), 184-206


Bernadette Faye and Florian Luca

On Y-coordinates of Pell equations which are members of a fixed binary recurrence

view    print

Published: February 12, 2020.
Keywords: Diophantine equations, binary recurrence, Pell equation.
Subject: 11D61,11B39,11D45.

In this paper, we show that if u is a fixed binary recurrent sequence of integers whose characteristic equation has real roots and (Xk,Yk) is the k-th solution of the Pell equation X2-dY2=1 for some non-square integer d>1, the equation Yku has at most two positive integer solutions k provided d exceeds some effectively computable number depending on u.


The authors thank the anonymous referee for pointing out to them reference [4] and Professor Mouhamed Moustapha Fall for useful discussions. Part of this work was done during a very enjoyable visit of F. L. at AIMS Sénégal in February 2018. This author thanks AIMS Sénégal for the hospitality and support. In addition, F. L. was supported in part by Grant CPRR160325161141 of NRF and the Number Theory Focus Area Grant of CoEMaSS at Wits (South Africa) and CGA 17-02804S (Czech Republic).

Author information

Bernadette Faye:
Université Gaston Berger
Saint-Louis 32002, Sénégal


Florian Luca:
School of Mathematics
University of the Witwatersrand
Private Bag X3, Wits 2050, Johannesburg, South Africa;
Research Group in Algebraic Structures and Applications
King Abdulaziz University, Jeddah, Saudi Arabia;
Max Planck Institute for Mathematics, Bonn, Germany;
Centro de Ciencias Matemáticas
UNAM, Morelia, Mexico