Almost perfect powers in consecutive integers (Q2717763)

Let \(P(n)\) be the greatest prime factor of a positive integer \(n\), with \(P(1)=1\). The authors consider equations of the type NEWLINE\[NEWLINE (n+d_1)\cdots (n+d_t) = b y^\ell \leqno(1) NEWLINE\]NEWLINE where \(b\), \(k\geq 2\), \(\ell\) odd prime, \(n>k^\ell\) and \(y\) are positive integers such that \(P(b)\leq k\), \(t=k-r\) with \(r\in \{0,1\}\), and \(d_1<\ldots <d_t\). Notice that if \(r=0\) then \(d_i=i\) for \(0\leq i <k\) whereas if \(r=1\) then the left-hand side of (1) is obtained by omitting a term \(n+i\) for some \(i\) with \(0\leq i <k\) from \(\{n,n+1,\ldots,n+k-1\}\). NEWLINENEWLINENEWLINEThe simplest example of such an equation is \((n+1)\cdots (n+k-1) = b y^\ell\). Erdős and Selfridge proved that it has no solution when \(P(b)<k\). NEWLINENEWLINENEWLINEHere the authors prove that equation (1) with \(r=1\) and \(k\in \{6,7,8\}\) has no solution and also when \(k\in \{3,5\}\) and \(P(b)<k\). Among many results the proof uses deep theorems of Darmon-Merel and Sander concerning generalizations of Fermat's equation.