# Article

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Keywords:
Lucas sequences; Diophantine equations; Pell equations
Summary:
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.
References:
[1] Muriefah, F.S. Abu, Rashed, A. Al: The simultaneous Diophantine equations \$y^{2}-5x^{2}=4\$ and \$z^{2}-442x^{2}=441\$. Arabian Journal for Science and Engineering, 31, 2, 2006, 207-211, MR 2284646
[2] Baker, A.: Linear forms in the logarithms of algebraic numbers (IV). Mathematika, 15, 2, 1968, 204-216, London Mathematical Society, DOI 10.1112/S0025579300002588 | MR 0258756
[3] Baker, A., Davenport, H.: The equations \$3x^{2}-2= y^{2}\$ and \$8x^{2}-7= z^{2}\$. Quart. J. Math. Oxford, 20, 1969, 129-137, MR 0248079
[4] Brown, E.: Sets in which \$xy+k\$ is always a square. Mathematics of Computation, 45, 172, 1985, 613--620, MR 0804949
[5] Cohn, J.H.E.: Lucas and Fibonacci numbers and some Diophantine equations. Glasgow Mathematical Journal, 7, 1, 1965, 24-28, Cambridge University Press, MR 0177944
[6] Copley, G.N.: Recurrence relations for solutions of Pell's equation. The American Mathematical Monthly, 66, 4, 1959, 288-290, JSTOR, DOI 10.2307/2309637 | MR 0103168
[7] Darmon, H., Granville, A.: On the equations \$z^{m}= f (x, y)\$ and \$ax^{p}+ by^{q}= cz^{r}\$. Bulletin of the London Mathematical Society, 27, 6, 1995, 513-543, Wiley Online Library, MR 1348707
[8] Grinstead, C.M.: On a method of solving a class of Diophantine equations. Mathematics of Computation, 32, 143, 1978, 936-940, DOI 10.1090/S0025-5718-1978-0491480-0 | MR 0491480
[9] Kedlaya, K.: Solving constrained Pell equations. Mathematics of Computation, 67, 222, 1998, 833-842, DOI 10.1090/S0025-5718-98-00918-1 | MR 1443123
[10] Mohanty, S.P., Ramasamy, A.M.S.: The simultaneous diophantine equations \$5y^{2}- 20= x^{2}\$ and \$2y^{2}+ 1= z^{2}\$. Journal of Number Theory, 18, 3, 1984, 356-359, Elsevier, MR 0746870
[11] Mordell, L.J.: Diophantine equations. Pure and Applied Mathematics, 30, 1969, Academic Press, MR 0249355
[12] Peker, B., Cenberci, S.: On the equations \$y^2-10x^2=9\$ and \$z^2-17x^2=16\$. International Mathematical Forum, 12, 15, 2017, 715-720, DOI 10.12988/imf.2017.7651
[13] Siegel, C.L.: Über einige Anwendungen diophantischer Approximationen. On Some Applications of Diophantine Approximations. Publications of the Scuola Normale Superiore, vol. 2, 2014, 81-138, Edizioni della Normale, Pisa, MR 3330350
[14] Szalay, L.: On the resolution of simultaneous Pell equations. Annales Mathematicae et Informaticae, 34, 2007, 77-87, MR 2385427
[15] Thue, A.: Über Ann{ä}herungswerte algebraischer Zahlen. Journal für die Reine und Angewandte Mathematik, 135, 1909, 284-305, MR 1580770

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