Problem 1139

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claims/: The 1 claim page of Problem 1139, one per claimant's result; the problem's standing derives from them.

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Statement. Let $1\leq u_1<u_2<\cdots$ be the sequence of integers with at most $2$ prime factors. Is it true that

$$\limsup \frac{u_{k+1}-u_k}{\log k}=\infty?$$

Formulation. Prime factors are counted with multiplicity, so the sequence consists of $1$, the primes and the products of two primes, equal or not. This is how the OEIS sequence A037143 that the site links and the formal-conjectures statement ($\Omega(n)\le2$) read the wording, and the claimed construction uses it, counting a forced square factor $p^2$ as two prime factors.

Status. The site labels the problem OPEN (page last edited 23 January 2026), with the explanation that the question cannot be settled by a finite computation. The site's proof-claims tab carries one entry, a full claim by Liam Price made with GPT Pro, as the tab gives it, first posted in the site's thread on 2026-06-19 and filed on the tab on 2026-07-15, not accepted by the site and recorded as pending on Price's claim page; it gives the derived standing claimed.

Source. erdosproblems.com/1139, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1139, https://www.erdosproblems.com/1139.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation (accessed 2026-09-04; thread accessed 2026-10-07) asks whether the gaps between consecutive integers with at most two prime factors exceed every constant multiple of $\log k$ infinitely often, $k$ the index; the site states the problem without commentary. Price's claim is the only proof claim, and its page records the route (a Chinese-remainder modulus making $N+n$ have at least three prime factors for every offset $n\le Y$, with the Green--Tao theorem on linear equations in primes behind the covering) and its standing: a thread comment reports a screening check with no issues found, which is not a review, and the site has not accepted it.

The thread also holds proposals that are not claims and have no claim page. A comment of 2026-01-24 reports that a literature search found no nontrivial lower bound for the gaps. Przemek Chojecki's posts of 26 to 28 January 2026 propose a route through Maynard's form of the Erdős--Rankin construction, run twice in the residue classes $1$ and $3$ modulo $4$; the post of 2026-01-28 links an undated and unsigned PDF note whose Theorem 1 rests on a covering lemma in those two classes (its Lemma 1) that the note asserts with a remark in place of a proof. Terence Tao found it unlikely that the lemma would be easy, since only half the primes are available for sieving, and the author replied that he would concentrate on proving it, so no complete proof was claimed and the note has no claim page. A covering-reduction framework posted on 2026-04-29 isolates a sparse multi-covering statement as the open technical point and claims no proof.

Search scope (2026-10-07). The site's problem page, its thread and its proof-claims tab, and the formal-conjectures statement file at the pinned commit; no literature survey of known results beyond them.