Problem 452

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claims/: The 1 claim page of Problem 452, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\subseteq [x,2x]$ such that $\omega(n)>\log\log n$ for all $n\in I$?

Status. Open, the site's label (page last edited 28 October 2025). One pending partial claim, White's weighted-sieve upper bound, bounds the longest run from above without determining its size, so the derived standing is open with no full claim.

Source. erdosproblems.com/452, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #452, https://www.erdosproblems.com/452.

References.

• [Er37] Erdős, Paul, Note on the number of prime divisors of integers. J. London Math. Soc. (1937), 308-314.

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation). With $\omega(n)$ the number of distinct prime factors of $n$, the size $L(x)$ of the largest interval $I\subseteq[x,2x]$ on which $\omega(n)>\log\log n$ throughout. The site labels the problem OPEN (page last edited 28 October 2025).

Standing. Open. One pending partial claim, White's report of 2026-07-26 (claim page), bounds $L(x)$ by $\exp((1+o(1))\log x/\log\log x)$; it is not refereed and names no reviewer, and it does not determine $L(x)$, so the derived standing stays open.

What is known. Erdős [Er37] proved that the integers with $\omega(n)>\log\log n$ have density $1/2$. The Chinese remainder theorem gives an interval with $|I|\ge(1+o(1))\log x/(\log\log x)^2$, and the site's commentary suggests that intervals of length $(\log x)^k$ might exist for every $k$. From above, a thread post of 4 July 2026 gives $L(x)\le x^{O(1/\sqrt{\log\log x})}$ by comparing the mean, variance and skewness of the count of prime factors below a power of the interval's length, and reports that Erdős and Graham knew no upper bound; White's report of 26 July 2026 improves this to $\exp((1+o(1))\log x/\log\log x)$ by a weighted sieve, crediting the argument to GPT-5.6 Sol; a thread post of 28 September 2026 proves $L(x)\le x^{(1+\varepsilon)/\log\log x}$ for every $\varepsilon>0$ by a Brun sieve, cites the report, tabulates $L(x)$ for $x$ from $10^6$ to $5.3\times10^8$ and conjectures $L(x)\asymp\log x$. Between $\log x/(\log\log x)^2$ and $x^{o(1)}$ the size is undetermined, and neither thread post is a dated manuscript, so neither has a claim page.

Search scope. The site's page, its two comments and its empty proof-claims tab, the report, the formal-conjectures statement file and the community database; no other literature search was made.

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.

• erdos_1937_note_number_prime_divisors_integers
• erdos_1937_note_number_prime_divisors_integers / main_theorem