The Diophantine equation \((ax^k-1)(by^k-1)=abz^k-1\) (Q2637188)

For the background on this paper see the related one by \textit{E. G. Goedhart} and \textit{H. G. Grundman}, J. Number Theory 154, 74--81 (2015; Zbl 1360.11059). Here the author proves that for \(a,b\) be positive integers, the equation \[ (ax^k-1)(by^k-1)=abz^k-1 \] has no solutions in integers \(x\), \(y\), \(z\) and \(k\) with \(| x| >1\), \(| y| >1\) and \(k\geq 4\).