OBJECTIVE AND SOURCE REVIEW
Erdős #382; independent attempt, 2026-09-23. If the largest prime factor $Q$ of an interval product $\prod_{m=u}^v m$ occurs with multiplicity at least two, must $v-u=v^{o(1)}$, and can these lengths be unbounded? I read all of
382.tex, including its smooth-number anchor and factorization example. I examine higher multiplicity and the exact logical scope of that reduction.
WORK: MULTIPLICITY-SENSITIVE ANCHOR BOUND
Put $L=v-u$. Whenever $Q>L$, the interval has a unique multiple $t$ of $Q$: two such multiples would differ by at least $Q$. If the total exponent of $Q$ in the interval product is $r$, this unique term satisfies $Q^r\mid t$. Hence \[
L<Q\le v^{1/r}.
\] Thus a block with exponent at least three satisfies a cube-root bound, and a sequence of such blocks with $r\to\infty$ satisfies $L=v^{o(1)}$. The last statement follows by fixing $\varepsilon>0$ and eventually having $1/r<\varepsilon$. The supplied anchor theorem obtains $Q>L$ by a Sylvester--Schur step. The displayed argument above is explicitly conditional on that inequality; it verifies the subsequent valuation step without treating the imported theorem as an in-file proof. In particular it shows that, within that framework, the hardest unresolved case has bounded multiplicity, especially exponent two.
ADVERSARIAL VERIFICATION AND REMAINING GAP
Total exponent need not belong to one term unless $Q>L$ is first proved. The anchor is a multiple of $Q^r$, not necessarily $Q^r$ itself. Conversely, if every interval term is $Q$-smooth and a term is divisible by $Q^2$ with $Q$ prime, then the largest prime factor really is $Q$ and the interval is bad. This converse requires the full smoothness condition; a square anchor alone proves nothing. No unbounded family of smooth blocks has been constructed here. For fixed $r=2$, the power $1/r$ does not tend to zero, so this refinement does not settle the original subpower assertion.
SOURCES CHECKED
Complete local source
gpt_pro_5.4/382.tex; official indexed statement checked at
https://www.erdosproblems.com/382, 2026-09-23. No claim is made to have rerun the source's large factorization search.
FINAL
UNRESOLVED. The anchor bound is refined by exact multiplicity, while bounded-multiplicity blocks and unbounded lengths remain unresolved.
COMPLETION ESTIMATE
Conservative subjective progress toward both original questions.
COMPLETION: 8%