Some remarks on the Diophantine equation \((x^ 2-1)(y^ 2-1)=(z^ 2-1)^ 2\). (English) Zbl 1039.11011

Al-Kadhi, Mohammed; Kihel, Omar

Proc. Japan Acad., Ser. A 77, No. 10, 155-156 (2001).

In 1963, A. Schinzel and W. Sierpinski [Elem. Math. 18, 132–133 (1963; Zbl 0126.07301)] found all the integral solutions of the equation \[ (x^2-1)(y^2-1)=(z^2-1)^2, 1<y<z<x\tag{1} \] with \(x-y=2z\). In 1989, the reviewer conjectured that if \(x-y=kz, k\in {\mathbb N}\) with \(k\neq 2\), then equation (1) has no integral solutions. Furthermore, the reviewer got a generalization of the Schinzel-Sierpinski’s result [see J. Harbin Inst. Technol. 23, 9–14 (1991; Zbl 0971.11503)]. For example, it is proved that if \(k\leq 30\) then the conjecture is correct. Y. Wang [J. Nat. Sci. Heilongjiang Univ. 1989, No. 4, 84–85 (1989; Zbl 0979.11507)] proved that the conjecture holds for \(k=1\) or \(31\).
In this paper, the authors prove that if \(k\neq 2\) then equation (1) has finitely many solutions satisfying \(x-y=kz\). Moreover, \(k<z< k^2/2,1<y<k^2/2\) and \(k^2+1<x<(k^3+k^2)/2\). From the result and computer, the authors also prove that if \(k\neq 2\) and \(1\leq k\leq 10^3\) then equation (1) has no solution.
In fact, M. Z. Garaev and V. N. Chubarikov [Math. Notes 66, No. 2, 142–147 (1999); translation from Mat. Zametki 66, No. 2, 181–187 (1999; Zbl 0978.11009)] proved that the reviewer’s conjecture is correct for any \(k\in{\mathbb N}\).

Reviewer: Zhenfu Cao (Shanghai)

MSC:

11D25

Cubic and quartic Diophantine equations

Keywords:

quartic Diophantine equation

Citations:

Zbl 0126.07301; Zbl 0971.11503; Zbl 0979.11507; Zbl 0978.11009

References:

[1]	Cao, Z.: A generalization of the Schinzel-Sierpiński system of equations. J. Harbin Inst. Tech., 23 (5), 9-14 (1991). · Zbl 0971.11503
[2]	Grelak, A.: On the Diophantine equation \((x^2-1)(y^2-1) = (z^2-1)^2\). Discuss. Math., 5 , 41-43 (1982). · Zbl 0507.10009
[3]	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). · Zbl 0126.07301
[4]	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).
[5]	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). · Zbl 0857.11009