OPEN

Is it true that for any $n,k\geq 1$, if $n+1,\ldots,n+k$ are all composite then there are distinct primes $p_1,\ldots,p_k$ such that $p_i\mid n+i$ for $1\leq i\leq k$?

Note this is trivial when $k\leq 2$. Originally conjectured by Grimm. This is a very difficult problem, since it in particular implies $p_{n+1}-p_n <p_n^{1/2-c}$ for some constant $c>0$, in particular resolving Legendre's conjecture.

Grimm proved that this is true if $k\ll \log n/\log\log n$. Erdős and Selfridge improved this to $k\leq (1+o(1))\log n$. Ramachandra, Shorey, and Tijdeman [RST75] have improved this to \[k\ll\left(\frac{\log n}{\log\log n}\right)^3.\]