Almost perfect powers in consecutive integers. II (Q735435)

Let \(k\) be an integer. The authors find all integers of the form \(by^\ell\), where \(\ell\geq2\), and the greatest prime factor of \(b\) is at most \(k\) such that they are all products of \(k\) consecutive integers with two terms omitted. Namely, if these integers are \(n+d_1\), \(n+d_2\), \dots, \(n+d_{k-2}\), \(0\leq d_1<d_2<d_{k-2}<k\), then the only solutions are obtained for \(\ell=3\), \(k=4\), \(n=125\) or \(k=7\) and the greatest prime factor of \(b\) is \(7\). This result can be compared to a previous result of the same authors (with \textit{G. Hanrot}) [Acta Arith. 99, No. 1, 13--25 (2001; Zbl 0971.11017)] where exactly one term was omitted. The proof involves the solution of difficult exponential and Thue equations.