Erdős Problem 682: rough integers in prime gaps
Statement and claim.
Let $p_n<p_{n+1}$ be consecutive primes. For almost all indices $n$, there is an integer $m$ strictly between them whose least prime factor is at least $p_{n+1}-p_n$. The main analytic input is the established theorem of Gafni and Tao,
Rough numbers between consecutive primes, Theorem 1.1:
https://arxiv.org/abs/2508.06463. Below we explain its precise counting parameter and derive the assertion, together with an elementary obstruction for other gaps. See
https://www.erdosproblems.com/682.
External theorem and change of parameter.
The cited theorem bounds the number of exceptional gaps with
prime value $p_n\in[X,2X]$ by \[
O\left(\frac X{(\log X)^2}\right). \tag{1}
\] This is an external analytic theorem; its sieve argument is not reproved here. It is important that $X$ in $1$ is not an index. Let $E(Y)$ count all exceptional gaps starting at a prime at most $Y$. Split $(\sqrt Y,Y]$ into dyadic intervals. On these intervals the logarithms are at least a constant times $\log Y$, and their lengths sum to at most $Y$. The remaining gaps starting below $\sqrt Y$ number at most $\sqrt Y$. Hence \[
E(Y)\ll \frac Y{(\log Y)^2}+\sqrt Y
\ll\frac Y{(\log Y)^2}. \tag{2}
\] By the prime number theorem, $p_N\sim N\log N$ and $\log p_N\sim\log N$. Consequently the number of exceptional indices $n\leq N$ is \[
E(p_N)\ll \frac N{\log N}=o(N). \tag{3}
\] This proves the required almost-all conclusion. An $O(N/(\log N)^2)$ bound with $N$ an index does not follow from $1$, and is not claimed in this reconstruction.
An explicit local obstruction.
Suppose $p>3$ and both $p,p+4$ are prime. Necessarily $p\equiv1\pmod6$: the other possible odd prime residue, $5\pmod6$, would make $p+4$ divisible by $3$. The three integers inside this gap then satisfy \[
2\mid p+1,\qquad 3\mid p+2,\qquad 2\mid p+3.
\] Thus $p,p+4$ are consecutive primes, and every interior integer has least prime factor strictly smaller than the gap length $4$. Every such cousin-prime gap is exceptional; for example the gap from $7$ to $11$ is exceptional. This does not prove that infinitely many exceptions exist, because infinitude of these prime pairs is an additional unproved input and is not being assumed. By contrast, every twin-prime gap has its even midpoint as a witness with least prime factor $2$, exactly its gap length.
Scope.
The density-one assertion is fully proved relative to the cited Gafni--Tao theorem and the prime number theorem. The parameter conversion and the local gap obstruction are established above. No assertion about the infinitude of exceptional gaps is needed.
Completion estimate.
COMPLETION: 100%