The Diophantine equation \((x^{k} - 1)(y^{k} - 1) = (z^{k} - 1)^{t}\)
The main results of this paper, which improve on \textit{Y. Bugeaud}'s [Indag. Math., New Ser. 15, No. 1, 21--28 (2004; Zbl 1098.11022)], are Theorem 1. The Diophantine equation \[ (x^k-1)(y^k-1)=(z^k-1) \] has only the solutions \[ (x,y,z,k)=(-1,4,-5,3)\quad \text{and}\quad (4,-1,-5,3)\; \] in integers \(x\), \(y\), \(z\), and \(k\) whith \(|z|\geq 2\) and \(k\geq3\). and Theorem 2. The Diophantine equation \[ (x^k-1)(y^k-1)=(z^k-1)^2 \] has no solutions in integers \(x\), \(y\), \(z\), and \(k\) with \(|x|\not=|y|\), \(|z|\geq 2\) and \(k\geq 4\). Notice that theorem 1 is best possible. The proofs use the hypergeometric method of Thue-Siegel, several gap principles and also the modular method.
scientific article; zbMATH DE number 5311046