Problem 380

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claims/: The 2 claim pages of Problem 380, one per claimant's result; the problem's standing derives from them.

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Statement. We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\leq x$ which are contained in at least one bad interval. Is it true that

$$B(x)\sim \#\{ n\leq x: P(n)^2\mid n\},$$

where $P(n)$ is the largest prime factor of $n$?

Status. Proved, the site's label (PROVED, page last edited 10 April 2026): Tao's 2026 preprint proves the asymptotic with a relative error of a power of the logarithm, and the site's curator records it as the proof. The accepted claim is Tao 2026; a Lean development that proves the asymptotic by its own route is the pending claim Alexeev 2026.

Source. erdosproblems.com/380, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #380, https://www.erdosproblems.com/380.

References.

• [Ta26c] T. Tao, Products of consecutive integers with unusual anatomy. arXiv:2603.27990 (2026).

Formalization. The site reports no formalised statement, and no formal-conjectures statement file is recorded for the problem. A Lean 4 development in the lean-proofs repository, Erdos380.lean at its commit of 2026-08-26, proves erdos380 : B ~[Filter.atTop] repeatedLargestPrimeCount, the two-sided asymptotic without an error term, by a route of its own; it is recorded as the claim Alexeev 2026 and has not been built or audited by this corpus.

Current assessment

Proved by Tao 2026, accepted by the site's curator. The site formulation above (page last edited 10 April 2026) asks whether the integers up to $x$ covered by a bad interval are asymptotically the integers $n\le x$ with $P(n)^2\mid n$, the singleton bad intervals; the site records that Erdős and Graham knew only $B(x)>x^{1-o(1)}$ and that the comparison count is $x/\exp((c+o(1))\sqrt{\log x\log\log x})$ for some $c>0$. Tao's preprint [Ta26c] (arXiv, 2026-03-30; third version 2026-09-26) proves $B(x)=(1+O((\log x)^{-1+o(1)})),#{n\le x:P(n)^2\mid n}$ (Theorem 1.7) by the Guth–Maynard zero-density estimate and the large sieve (built on Montgomery's uncertainty lemma) for the long bad intervals and an anti-sieve with character-sum moment bounds for the short ones, and the site's curator labels the problem PROVED on that preprint, which is the accepted claim's reviewed evidence; no journal publication is recorded. The same preprint settles the site's companion remark: the integers in very bad intervals (product powerful) that are not themselves powerful number $O(x^{2/5+o(1)})$ (Theorem 1.8), so their count is asymptotic to the powerful numbers, $(\zeta(3/2)/\zeta(3))\sqrt x$. The forum thread (ten comments, 2025-09-13 to 2026-04-01) carries Tao's earlier observations, which the site's commentary reports: a bad interval contains no prime, so $v<2u$ by Bertrand's postulate and, under Cramér's conjecture, $v-u\ll(\log u)^2$; the Erdős–Graham remark $B(x)>x^{1-o(1)}$ is puzzling since $B(x)$ trivially dominates the singleton count, and Tao suggests from Erdős and Graham's 1976 paper on products of factorials that $B(x)=o(x)$ was meant; the constant $c$ can be taken as $\sqrt2$. Tao's announcement of 2026-03-31 adds that the Guth–Maynard estimate plays a crucial role and that the older zero-density estimates just fail. Lean: the site reports no formal-conjectures statement; the lean-proofs development of 2026-08-26 proves the qualitative asymptotic by its own route (the pending claim Alexeev 2026), declares no informal author, and has not been built or audited by this corpus, so no formalized evidence is listed on either page. The community database's commit of 2026-03-31 changed the problem's status to proved, and the database lists it as unformalized (as of 2026-10-06).

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.

• luca_2014_squares_factorials_products_factorials
• luca_2014_squares_factorials_products_factorials / theorem_2
• tao_2026_products_consecutive_integers_unusual_anatomy
• tao_2026_products_consecutive_integers_unusual_anatomy / lemma_6_1
• tao_2026_products_consecutive_integers_unusual_anatomy / theorem_1_7
• tao_2026_products_consecutive_integers_unusual_anatomy / theorem_1_8