OPEN
Is it true that for almost all $n$ there exists some $m\in (p_n,p_{n+1})$ such that
\[p(m) \geq p_{n+1}-p_n,\]
where $p(m)$ denotes the least prime factor of $m$?
Erdős first thought this should be true for all large $n$, but found a (conditional) counterexample:
Dickson's conjecture says there are infinitely many $d$ such that
\[2183+30030d\textrm{ and }2201+30030d\]
are both prime, and then they must necessarily be consecutive primes. These give a counterexample since $30030=2\cdot 3 \cdot 5\cdot 7\cdot 11\cdot 13$ and every integer in $[2184,2200]$ is divisible by at least one of these primes.
See also [680] and [681].
