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Claim. The note Truncated Congruence Sieves and Erdős Problem 25 (19 March 2026) writes $B_i=\{n\in\mathbb N:n\ge n_i,\ n\equiv a_i\pmod{n_i}\}$ and $A=\mathbb N\setminus\bigcup_iB_i$, the set of Problem 25, and $A^{(k)}=\mathbb N\setminus\bigcup_{i\le k}B_i$ for its finite truncations, which are eventually periodic with densities $\delta_k$ decreasing to a limit $\delta$ (Lemma 2.1). Its Theorem 3.1 (pp. 3--4) states that if $\sum_i1/n_i<\infty$ then $A$ has natural density, equal to $\delta$, and hence logarithmic density: the tail bound $\overline d(A^{(k)}\setminus A)\le\sum_{i>k}1/n_i$ squeezes the upper and lower densities of $A$ between $\delta_k-\sum_{i>k}1/n_i$ and $\delta_k$. Its Theorem 3.2 (p. 4) states that if the $n_i$ are pairwise coprime then $A$ has natural density in all cases: the Chinese remainder theorem gives $\delta_k=\prod_{i\le k}(1-1/n_i)$, Theorem 3.1 applies when $\sum_i1/n_i$ converges, and when it diverges the product, and with it $d(A)$, is $0$. Each theorem answers the problem's question yes for the sequences it covers. The note's statements are recorded on its card chojecki_2026_truncated_congruence_sieves_erdos_problem_25; no step of its proofs has been checked independently.

Submission note. Posted to the site's forum by Przemyslaw Chojecki on 19 March 2026:

After extensive back-and-forth with GPT-5.4 Pro I've managed to get 2 unconditional proofs for special cases and then a general strategy with identified obstacles (quotient sieves). Seems like the full problem resolution is a question of time, when these sieves get better. Here's the full note.

Covers. Every sequence of moduli with $\sum_i1/n_i<\infty$, and every sequence of pairwise coprime moduli, with arbitrary residue classes in both cases; not the general problem, for which the same note records a reduction to an unproved estimate on [[problems/integer_sequences/E0025/claims/2026_03_19_chojecki_conditional|its conditional page]].

Standing. Przemek Chojecki posted the note in the site's thread on 19 March 2026, writing that they had reached two unconditional proofs for special cases and a general strategy with identified obstacles after extended work with GPT-5.4 Pro; Chojecki is the claimant as its submitter, and GPT-5.4 Pro is the system they name. The note has no refereed publication, no formalization and no outside review, and the site's label is unchanged (OPEN) and credits the note with nothing. Nat Sothanaphan replied the same day that a standard check, a linked ChatGPT conversation, claims one minor issue; that is a screening report, not a review. The claim stays claimed.

Depends on. Nothing on the wiki.