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Claim. There is an absolute constant $C>0$ such that infinitely many $n$ satisfy $\omega(n+k)\le Ck$ for every integer $k\ge1$. This is Theorem 1.1 of the preprint, which proves the bound for $\Omega(n+k)$, the number of prime factors counted with multiplicity, and hence for $\omega(n+k)\le\Omega(n+k)$. It answers the question of Problem 248 yes, with the implied constant in $\omega(n+k)\ll k$ absolute. The proof runs a Maynard-type high-dimensional sieve whose dimension grows slowly and handles the shifts $k\gg\log\log n$ inside the sieve weights rather than by counting bounds; the authors' forum announcement of 2025-12-01 explains that the shifts $k=o(\log\log n)$ carry the difficulty. The source card is Tao and Teräväinen 2025.

Acceptance. The site's curator, Thomas F. Bloom, records in the problem's commentary (page last edited 2026-04-17) that the problem has been resolved by this result and labels the problem proved; that documented acceptance is the reviewed evidence. The preprint (arXiv:2512.01739, v1 of 2025-12-01, v2 of 2026-04-25) has no journal publication on record, so refereed is not listed. A Lean 4 development in the lean-proofs repository (file first published 2026-08-23) declares itself a formalization of a solution to the problem with Tao and Teräväinen as its informal authors and proves the same statement as Erdos248.erdos_248; its header credits the formal proof to the AI systems Codex and GPT-5.6 Sol and the repository to Boris Alexeev, and says that it implements the Tao–Teräväinen weighted-sieve argument directly for $\omega$. The formal-conjectures project links that file as the formal proof of its statement erdos_248 (category research solved), which is the Lean that the site's label refers to; the statement file is linked above as a record, since it holds no proof. This corpus has not built the development, printed its axioms or audited its statement, so formalized is not listed.

Later strengthening. Lau (arXiv:2604.15042, 2026-04-16) proves that for some absolute $C$ infinitely many $n$ have $\Omega(n+k)\le C\log k$ for every $k\ge2$, which the site records as an improvement of this bound; see Lau 2026. It refines the method of this claim, and after the shift $n\mapsto n+1$ it implies the problem's statement by itself, since $\omega(n+1+j)\le\Omega(n+(j+1))\le C\log(j+1)\le Cj$ for every $j\ge1$; it is recorded as a second accepted claim on Lau 2026.

Depends on. Nothing in this wiki; the argument is self-contained in the preprint.