Problem 929

Erdős Problem #929: declared status 'open'. Formalized: no. 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 $S(k)\geq k^{1-o(1)}$? Current best: The best bound on large gaps between primes due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18] (see [4]) implies$$S(k) \ll k \frac{\log\log\log k}{\log\log k\log\log\log\log k}.$$ References [FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. Prize: no. Tags: number theory.