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Solutions of the Diophantine Equation 7X2 + Y7 = Z2 from Recurrence Sequences

Solutions of the Diophantine Equation 7X2 + Y7 = Z2 from Recurrence Sequences AbstractConsider the system x2 − ay2 = b, P (x, y) = z2, 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 7X2 + Y 7 = Z2 if (X, Y) = (Ln, Fn) (or (X, Y) = (Fn, Ln)) where {Fn} and {Ln} represent the sequences of Fibonacci numbers and Lucas numbers respectively. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematics de Gruyter

Solutions of the Diophantine Equation 7X2 + Y7 = Z2 from Recurrence Sequences

Communications in Mathematics , Volume 28 (1): 12 – Jun 1, 2020

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Publisher
de Gruyter
Copyright
© 2020 Hayder R. Hashim, published by Sciendo
eISSN
2336-1298
DOI
10.2478/cm-2020-0005
Publisher site
See Article on Publisher Site

Abstract

AbstractConsider the system x2 − ay2 = b, P (x, y) = z2, 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 7X2 + Y 7 = Z2 if (X, Y) = (Ln, Fn) (or (X, Y) = (Fn, Ln)) where {Fn} and {Ln} represent the sequences of Fibonacci numbers and Lucas numbers respectively.

Journal

Communications in Mathematicsde Gruyter

Published: Jun 1, 2020

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