Proceedings of the Japan Academy, Series A, Mathematical Sciences

Some remarks on the diophantine equation $(x^2 - 1)(y^2 - 1) = (z^2 - 1)^2$

Mohammed Al-Kadhi and Omar Kihel

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Abstract

In this paper, among other results, we prove that the title equation has finitely many solutions when $x - y = lz$ and $l$ is a fixed integer $\ne 2$. Moreover, all solutions $(x, y, z)$ satisfy $l < z < l^2 / 2$, $1 < y < l^2 / 2$ and $l^2+1 < x < (l^3 + l^2) / 2$. As a consequence, we extend a result of Cao.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 10 (2001), 155-156.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148392979

Digital Object Identifier
doi:10.3792/pjaa.77.155

Mathematical Reviews number (MathSciNet)
MR1873734

Zentralblatt MATH identifier
1039.11011

Subjects
Primary: 11D09: Quadratic and bilinear equations 11D25: Cubic and quartic equations

Keywords
Diophantine equation

Citation

Al-Kadhi, Mohammed; Kihel, Omar. Some remarks on the diophantine equation $(x^2 - 1)(y^2 - 1) = (z^2 - 1)^2$. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 10, 155--156. doi:10.3792/pjaa.77.155. https://projecteuclid.org/euclid.pja/1148392979


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References

  • Cao, Z.: A generalization of the Schinzel-Sierpiński system of equations. J. Harbin Inst. Tech., 23(5), 9–14 (1991).
  • Grelak, A.: On the Diophantine equation $(x^2-1)(y^2-1) = (z^2-1)^2$. Discuss. Math., 5, 41–43 (1982).
  • Schinzel, A., et Sierpiński, W.: Sur l'équation diophantienne $(x^2-1)(y^2-1) = [((y-x)/2)^2-1]^2$. Elem. Math., 18, 132–133 (1963).
  • Wang, Y.: On the Diophantine equation $(x^2-1)(y^2-1) = (z^2-1)^2$. Heilongjiang Daxue Ziran Kexue Xuebao, 4, 84–85 (1989).
  • Wu, H., and Le, M.: A note on the Diophantine equation $(x^2-1)(y^2-1) = (z^2-1)^2$. Colloq. Math., 71(1), 133–136 (1996).