Problem Statement
Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.
Categories:
Number Theory
Progress
In other words, how large can a consecutive set of $k$-smooth integers be? Sylvester and Schur (see [Er34]) proved $f(k)\leq k$ and Erdős [Er55d] proved\[f(k)<3\frac{k}{\log k}.\]Jutila [Ju74] and Ramachandra, and Shorey [RaSh73] proved\[f(k) \ll \frac{\log\log\log k}{\log \log k}\frac{k}{\log k}.\]It is likely that $f(k) \ll (\log k)^{O(1)}$.This is essentially equivalent to [683].
Source: erdosproblems.com/961 | Last verified: January 19, 2026