Problem 375

Erdős Problem #375 [status: falsifiable; formalized: yes]. 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 $1\leq i\leq k$? Current best: Note this is trivial when $k\leq 2$. Grimm proved that this is true if $k\ll \log n/\log\log n$. Erd\H{o}s and Selfridge improved this to $k\leq (1+o(1))\log n$. Ramachandra, Shorey, and Tijdeman [RST75] have improved this to$$k\ll\left(\frac{\log n}{\log\log n}\right)^3.$$This is problem B32 in Guy's collection [Gu04]. Prize: no. Tags: number theory.