Solutions of the Diophantine Equation 7 X 2 + Y 7 = Z 2 from Recurrence Sequences

Hayder R. Hashim

Communications in Mathematics (2020)

  • Volume: 28, Issue: 1, page 55-66
  • ISSN: 1804-1388

Abstract

top
Consider the system x 2 - a y 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 n ) (or ( X , Y ) = ( F n , L n ) ) where { F n } and { L n } represent the sequences of Fibonacci numbers and Lucas numbers respectively.

How to cite

top

Hashim, Hayder R.. "Solutions of the Diophantine Equation $7X^2+Y^7=Z^2$ from Recurrence Sequences." Communications in Mathematics 28.1 (2020): 55-66. <http://eudml.org/doc/297308>.

@article{Hashim2020,
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 $7X^2+Y^7=Z^2$ if $(X,Y)=(L_n,F_n)$ (or $(X,Y)=(F_n,L_n)$) where $\lbrace F_n\rbrace $ and $\lbrace L_n\rbrace $ represent the sequences of Fibonacci numbers and Lucas numbers respectively.},
author = {Hashim, Hayder R.},
journal = {Communications in Mathematics},
keywords = {Lucas sequences; Diophantine equations; Pell equations},
language = {eng},
number = {1},
pages = {55-66},
publisher = {University of Ostrava},
title = {Solutions of the Diophantine Equation $7X^2+Y^7=Z^2$ from Recurrence Sequences},
url = {http://eudml.org/doc/297308},
volume = {28},
year = {2020},
}

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
PB - University of Ostrava
VL - 28
IS - 1
SP - 55
EP - 66
AB - 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 $\lbrace F_n\rbrace $ and $\lbrace L_n\rbrace $ represent the sequences of Fibonacci numbers and Lucas numbers respectively.
LA - eng
KW - Lucas sequences; Diophantine equations; Pell equations
UR - http://eudml.org/doc/297308
ER -

References

top
  1. Muriefah, F.S. Abu, Rashed, A. Al, The simultaneous Diophantine equations y 2 - 5 x 2 = 4 and z 2 - 442 x 2 = 441 , Arabian Journal for Science and Engineering, 31, 2, 2006, 207-211, (2006) MR2284646
  2. Baker, A., 10.1112/S0025579300002588, Mathematika, 15, 2, 1968, 204-216, London Mathematical Society, (1968) MR0258756DOI10.1112/S0025579300002588
  3. Baker, A., Davenport, H., The equations 3 x 2 - 2 = y 2 and 8 x 2 - 7 = z 2 , Quart. J. Math. Oxford, 20, 1969, 129-137, (1969) MR0248079
  4. Brown, E., Sets in which x y + k is always a square, Mathematics of Computation, 45, 172, 1985, 613--620, (1985) MR0804949
  5. Cohn, J.H.E., Lucas and Fibonacci numbers and some Diophantine equations, Glasgow Mathematical Journal, 7, 1, 1965, 24-28, Cambridge University Press, (1965) MR0177944
  6. Copley, G.N., 10.2307/2309637, The American Mathematical Monthly, 66, 4, 1959, 288-290, JSTOR, (1959) MR0103168DOI10.2307/2309637
  7. Darmon, H., Granville, A., On the equations z m = f ( x , y ) and a x p + b y q = c z r , Bulletin of the London Mathematical Society, 27, 6, 1995, 513-543, Wiley Online Library, (1995) MR1348707
  8. Grinstead, C.M., 10.1090/S0025-5718-1978-0491480-0, Mathematics of Computation, 32, 143, 1978, 936-940, (1978) MR0491480DOI10.1090/S0025-5718-1978-0491480-0
  9. Kedlaya, K., 10.1090/S0025-5718-98-00918-1, Mathematics of Computation, 67, 222, 1998, 833-842, (1998) MR1443123DOI10.1090/S0025-5718-98-00918-1
  10. Mohanty, S.P., Ramasamy, A.M.S., The simultaneous diophantine equations 5 y 2 - 20 = x 2 and 2 y 2 + 1 = z 2 , Journal of Number Theory, 18, 3, 1984, 356-359, Elsevier, (1984) MR0746870
  11. Mordell, L.J., Diophantine equations, Pure and Applied Mathematics, 30, 1969, Academic Press, (1969) MR0249355
  12. Peker, B., Cenberci, S., 10.12988/imf.2017.7651, International Mathematical Forum, 12, 15, 2017, 715-720, (2017) DOI10.12988/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, (2014) MR3330350
  14. Szalay, L., On the resolution of simultaneous Pell equations, Annales Mathematicae et Informaticae, 34, 2007, 77-87, (2007) MR2385427
  15. Thue, A., Über Ann{ä}herungswerte algebraischer Zahlen, Journal für die Reine und Angewandte Mathematik, 135, 1909, 284-305, (1909) MR1580770

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.