Problem 382

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Statement. Let $u\leq v$ be such that the largest prime dividing $\prod_{u\leq m\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large?

Status. Open, the site's label (OPEN).

Source. erdosproblems.com/382, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #382, https://www.erdosproblems.com/382.

Formalization. None recorded.

Current assessment

Open; the first question is answered only under a hypothesis. The site's formulation (its revision of 20 October 2025) asks, for intervals $[u,v]$ whose product has its largest prime factor to an exponent at least $2$, whether $v-u=v^{o(1)}$ and whether $v-u$ can be arbitrarily large; the site labels the problem OPEN. Erdős and Graham report that results of Ramachandra give $v-u\le v^{1/2+o(1)}$, and Remark 6.2 of Tao's preprint Tao 2026 says that Ramachandra's Selberg-sieve argument gives $v-u\ll v^{1/2-c}$ for an absolute $c>0$, records both questions as conjectures of Erdős and Graham and makes no progress on them. The site's commentary credits Stijn Cambie with the observation that a consequence of Cramér's conjecture, that every interval $[u,u+u^{\epsilon}]$ with $u$ large contains a prime, answers the first question yes, since an interval that contains a prime has its largest prime factor to the first power only. That answer rests on an unproved hypothesis and is a remark in the site's commentary, not a dated manuscript, so it gets no claim page. Cambie's heuristic for the second question is not a proof; a positive answer to Problem 383 would answer that question yes. A comment of 11 August 2025 in the problem's forum thread gives an interval with $v-u=13$ whose largest prime factor $4237033$ appears squared and one with $v-u=5$ whose largest prime factor $211193$ appears cubed.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• tao_2026_products_consecutive_integers_unusual_anatomy
• tao_2026_products_consecutive_integers_unusual_anatomy / lemma_6_1