Problem 369

Erdős Problem #369: declared status 'proved'. Formalized: no. 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? Current best: If this is the problem then the answer is yes, which follows from a result of Balog and Wooley [BaWo98]: for any $\epsilon>0$ and $k\geq 2$ there exist infinitely many $m$ such that $m+1,\ldots,m+k$ are all $m^\epsilon$-smooth.{/LI} {LI}Each $m\in P$ must be in $[n/2,n]$ (say). Prize: no. Tags: number theory.