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## Abstract

Consider the system *x*
^{2} − *ay*
^{2} = *b*, *P* (*x, y*) = *z*
^{2}, where *P* is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7*X*
^{2} + *Y*
^{7} = *Z*
^{2} if (*X, Y*) = (*L*
* _{n}*,

*F*

*) (or (*

_{n}*X, Y*) = (

*F*

*,*

_{n}*L*

*)) where {*

_{n}*F*

*} and {*

_{n}*L*

*} represent the sequences of Fibonacci numbers and Lucas numbers respectively.*

_{n}