Problem 680

Erdős Problem #680: declared status 'open'. Formalized: yes. Is it true that, for all sufficiently large $n$, there exists some $k$ such that$$p(n+k)>k^2+1,$$where $p(m)$ denotes the least prime factor of $m$? Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant? Prize: no. Tags: number theory, primes.