Erdős Problems

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OPEN

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 $S(k)\geq k^{1-o(1)}$?

#929: [Er76d]

number theory

It follows from Rosser's sieve that $S(k)> k^{1/2-o(1)}$.

It is trivial that $S(k)\leq k$ since, for example, one can take $n\equiv 1\pmod{k!}$. The best bound on large gaps between primes due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18] (see [4]) implies \[S(k) \ll k \frac{\log\log\log k}{\log\log k\log\log\log\log k}.\]