Erdős Problem 457
Statement and result.
Are there $\epsilon>0$ and infinitely many integers $n$ for which every prime $p\leq(2+\epsilon)\log n$ divides $\prod_{i=1}^{\lfloor\log n\rfloor}(n+i)$? Yes. In fact any fixed constant can replace $2+\epsilon$. We reconstruct the simultaneous-approximation method described by Barreto and Tao in the current discussion.
The elementary approximation lemma.
For real numbers $\alpha_1,\ldots,\alpha_s$ and integer $Q\geq2$, there exists $1\leq t\leq Q^s$ such that $\|t\alpha_j\|\leq1/Q$ for every $j$. Partition $[0,1)^s$ into $Q^s$ half-open cubes of side $1/Q$; two of the $Q^s+1$ points with coordinates $\{r\alpha_j\}$, $0\leq r\leq Q^s$, lie in the same cube. Subtracting their indices proves the lemma.
Construct a long enough interval without making its start too large.
Fix $A>2$, let $k$ tend to infinity through integers, and set \[
B=\lceil e^k\rceil,\qquad Q=\lceil4A\rceil.
\] Apply the lemma to the numbers $B/p$, for primes $k-1<p\leq Ak$. If $s$ is the number of these primes, choose the resulting $t$ and put \[
z=Bt,\qquad n=z-\lfloor k/2\rfloor.
\] For each such $p$ there is an integer multiple $m_p p$ with \[
|z-m_pp|\leq p/Q\leq k/4.
\] Therefore, for sufficiently large $k$, \[
1\leq m_pp-n\leq k-1.
\tag{1}
\] Also $n\geq e^k-k/2$, so $\lfloor\log n\rfloor\geq k-1$ for large $k$. Equation $1$ puts a multiple of every prime in $(k-1,Ak]$ into the required interval. Every prime $p\leq k-1$ already divides any product of $k-1$ consecutive integers. Hence every prime $p\leq Ak$ divides that product.
The size comparison and infinitely many starts.
The standard Chebyshev bound $\pi(x)=O(x/\log x)$, with $A$ fixed, gives $s=O_A(k/\log k)$. Since $1\leq t\leq Q^s$, \[
\log n\leq\log B+s\log Q
=k+O_A(k/\log k),
\] while the preceding lower bound gives $\log n=k+O_A(k/\log k)$. For any fixed $C<A$, eventually $C\log n<Ak$. Thus all primes $p\leq C\log n$ divide $\prod_{i=1}^{\lfloor\log n\rfloor}(n+i)$. The starts $n\geq e^k-k/2$ are unbounded, so infinitely many distinct starts occur. Taking, for example, $C=3$ proves the question.
Sources and scope.
https://www.erdosproblems.com/457 and
https://www.erdosproblems.com/forum/thread/457, checked 24 September 2026, attribute the constructions to GPT-5.2 Pro prompted by Kevin Barreto, with a growing-constant refinement explained by Terence Tao and expanded by Nat Sothanaphan. Here the approximation lemma is proved; Chebyshev's prime-counting bound is external. The stronger growing-constant rate and the separate upper-bound Problem 1181 are outside this reconstruction.
Completion estimate.
The literal question is proved, in the stronger form allowing any fixed constant.
COMPLETION: 100%