OPEN
This is open, and cannot be resolved with a finite computation.
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 $f(k)$.
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].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #961, https://www.erdosproblems.com/961, accessed 2026-01-17
