For positive integers \(n\) and \(k\), put \(\Delta(n,k)=n(n+1)\dots (n+k-1)\). In the paper under review, the authors extend results of Erdős and Selfridge, Saradha, and Saradha and Shorey concerning the squarefree part of \(\Delta(n,k)\). Write \(G\) for the set of \(i\in \{0,1,\dots,k-1\}\) such that \(n+i\) is divisible by a prime \(>k\) at an odd power, and put \(g\) for the cardinality of \(G\). The authors Theorem 1 shows that if \(k\geq 10\) and \(n>k^2\), then \(g\geq 8\), except for finitely many pairs \((n,k)\) all of which are given. As a corollary, if \(n>k^2\), \(k\geq 14\), or \(k\geq 10\) but \(n\geq 5040\), then there are at least \(8\) distinct primes dividing \(\Delta(n,k)\) at an odd power. The authors derive their Theorem 1 from Theorem 2, a more general statement which also covers the case \(k<10\). The authors also prove a finiteness result (Theorem 3) concerning squares of the form \((n+d_1)\dots (n+d_{k-t})\), where \(t\in [2,7]\), \(d_1,\dots,d_{k-t}\) are distinct integers in \(\{0,\dots,k-1\}\) and \(n>k^2\). In particular, if \(k\geq 4\), then the product of \(k-2\) distinct terms out of \(k\) consecutive positive integers is a square only in finitely many instances all of which are given.
The proofs use combinatorial arguments to reduce all such equations to the determination of all the integer points on several elliptic curves. This last task is achieved by using the computer package SIMATH.