Some remarks on the Diophantine equation \((x^ 2-1)(y^ 2-1)=(z^ 2-1)^ 2\).
In 1963, \textit{A. Schinzel} and \textit{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. \textit{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, \textit{M. Z. Garaev} and \textit{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 \(k\in{\mathbb N}\).
scientific article; zbMATH DE number 1782971