Perfect powers in products of arithmetical progressions with fixed initial term (Q675791)

The author considers the exponential diophantine equation \[ m(m+d)\cdots(m+ (k-1)d)= by^\ell\tag{\(*\)} \] in positive integers \(b\), \(d\), \(k\), \(\ell\), \(m\) satisfying \(P(b)\leq k\), \(\text{gcd}(m,d)= 1\), \(k>2\) and \(\ell\geq 2\). As usual, \(P(x)\) denotes the greatest prime factor of \(x\). The history of this equation goes back to Fermat, and many results have been obtained since. In the list of contributors both the author and his many times over joint author R. Tijdeman occupy prominent positions. P. Erdös conjectured that \((*)\) implies that \(k\) is bounded by an absolute constant. In the present paper, this conjecture is confirmed when \(\ell\geq 7\) and \(m\) fixed. This is an immediate consequence of the following elegant result: If \(\ell\geq 7\) in \((*)\) then \[ m\geq k^{C\log\log k}, \] where \(C>0\) is an effectively computable absolute constant.