Problem #682: Is it true that for almost all $n$ there exists some $m\in...

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Progress

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].