V. G. Sprindzhuk's problem on the squarefree part of the product of natural numbers (Q1280311)

Let \(Q(a)\) be the square-free part of the number \(a\). In the monograph by \textit{V. G. Sprindzhuk} [Classical diophantine equations in two unknowns, Moscow (1982; Zbl 0523.10008)] a problem is posed: does there exist a constant \(c\geq 1\) such that for infinitely many pairs of natural integers \(n\) and \(k\) with the condition \(k<(\ln n)^c\) the inequality \[ Q((n+1)\dots(n+k))< k^k \] is satisfied. The author proves the following general result: there exist positive constants \(c_5\), \(c_6\) and \(c_7\) such that for \(n\geq c_5\) \[ \begin{aligned} &Q((n+1)\dots(n+k)) > k^k,\qquad 1\leq k\leq c_6n^{1/2},\\ &Q((n+1)\dots(n+k))< k^k,\qquad k\geq c_7n^{1/2}.\end{aligned} \] {}.