Erdős Problem #369

Let \(\epsilon>0\) and \(k\geq 2\). Is it true that, for all sufficiently large \(n\), there is a sequence of \(k\) consecutive integers in \(\{1,\ldots,n\}\) all of which are \(n^\epsilon\)-smooth? The problem is trivially true as written (simply taking \(\{1,\ldots,k\}\) and \(n>k^{1/\epsilon}\)). There are (at least) two possible variants which are non-trivial, and it is not clear which Erdős and Graham meant. We formalize the second: each \(m\in P\) (where \(P\) is the sequence of \(k\) consecutive integers sought for) must be in \([n/2,n]\). In this case a positive answer also follows directly from the result of Balog and Wooley [BaWo98] for infinitely many \(n\). Proving this is true for all large \(n\) does not follow immediately from [BaWo98], but can be deduced using a similar construction, as shown by SkyYang.