PROVED
This has been solved in the affirmative.
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.
This was solved in the affirmative by Gafni and Tao
[GaTa25], who proved that the number of exceptional $n\in [1,X]$ is\[\ll \frac{X}{(\log X)^2},\]and proved, conditional on a form of the prime tuples conjecture, that the number of exceptional $n\in[1,X]$ satisifes\[\sim c\frac{X}{(\log X)^2}\]for some explicit $c>0$.
See also
[680] and
[681].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #682, https://www.erdosproblems.com/682, accessed 2026-04-11
