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

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 
@article{COMIM_2020__28_1_a4,
     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/}
}
TY  - JOUR
AU  - Hashim, Hayder R.
TI  - Solutions of the Diophantine Equation $7X^2+Y^7=Z^2$ from Recurrence Sequences
JO  - Communications in Mathematics
PY  - 2020
SP  - 55
EP  - 66
VL  - 28
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/COMIM_2020__28_1_a4/
LA  - en
ID  - COMIM_2020__28_1_a4
ER  - 
%0 Journal Article
%A Hashim, Hayder R.
%T Solutions of the Diophantine Equation $7X^2+Y^7=Z^2$ from Recurrence Sequences
%J Communications in Mathematics
%D 2020
%P 55-66
%V 28
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/COMIM_2020__28_1_a4/
%G en
%F COMIM_2020__28_1_a4
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/