Problem Statement
Let $k\geq 2$ be large and let $S(k)$ be the minimal $x$ such that there is a positive density set of $n$ where\[n+1,n+2,\ldots,n+k\]are all divisible by primes $\leq x$.
Estimate $S(k)$ - in particular, is it true that $S(k)\geq k^{1-o(1)}$?
Estimate $S(k)$ - in particular, is it true that $S(k)\geq k^{1-o(1)}$?
Categories:
Number Theory
Progress
It follows from Rosser's sieve that $S(k)> k^{1/2-o(1)}$.It is trivial that $S(k)\leq k+1$ since, for example, one can take $n\equiv 1\pmod{(k+1)!}$. The best bound on large gaps between primes due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18] (see [4]) implies\[S(k) \ll k \frac{\log\log\log k}{\log\log k\log\log\log\log k}.\]
Source: erdosproblems.com/929 | Last verified: January 19, 2026