Geodesic
Solutions of the Diophantine Equation $7X^2+Y^7=Z^2$ from Recurrence Sequences
Communications in Mathematics, Tome 28 (2020) no. 1, pp. 55-66.

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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 $7X^2+Y^7=Z^2$ if $(X,Y)=(L_n,F_n)$ (or $(X,Y)=(F_n,L_n)$) where $\{F_n\}$ and $\{L_n\}$ represent the sequences of Fibonacci numbers and Lucas numbers respectively.
Classification : 11B39, 11D41
Keywords: Lucas sequences; Diophantine equations; Pell equations 
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     author = {Hashim, Hayder R.},
     title = {Solutions of the {Diophantine} {Equation} $7X^2+Y^7=Z^2$ from {Recurrence} {Sequences}},
     journal = {Communications in Mathematics},
     pages = {55--66},
     publisher = {mathdoc},
     volume = {28},
     number = {1},
     year = {2020},
     mrnumber = {4124290},
     zbl = {07368973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2020__28_1_a4/}
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Hashim, Hayder R. Solutions of the Diophantine Equation $7X^2+Y^7=Z^2$ from Recurrence Sequences. Communications in Mathematics, Tome 28 (2020) no. 1, pp. 55-66. http://geodesic.mathdoc.fr/item/COMIM_2020__28_1_a4/