---

Claim. Theorem 5.4 (pp. 7--9) of the note Truncated Congruence Sieves and Erdős Problem 25 (19 March 2026) states that, assuming the note's Conjecture 5.1, the logarithmic density of the set $A$ of Problem 25 exists and equals $\delta=\lim_k\delta_k$, the limit of the densities $\delta_k$ of the finite truncations $A^{(k)}=\mathbb N\setminus\bigcup_{i\le k}B_i$, where $B_i=\{n\ge n_i:n\equiv a_i\pmod{n_i}\}$. The argument splits $\mathbb N\setminus A$ into the first-kill sets $E_i=A^{(i-1)}\cap B_i$, writes each as $\{a_i+n_it:t\in S_i\}$ for a finite periodic quotient sieve $S_i$ of density $d_i$ (Propositions 4.1 and 4.2), and sums the harmonic masses of the $E_i$ up to $X$, where Lemma 5.2 and Corollary 5.3 keep the total of the entropy terms $d_i\log(2/d_i)$ at $O(\log\log X)$. 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.

Hypothesis. Conjecture 5.1 of the note: with $\alpha_i=a_i/n_i$, there are nonnegative charges $\tau_i$ such that, for every $i$ and every $Y\ge1$,

$$\sum_{\substack{t\le Y\\ t\in S_i}}\frac1{t+\alpha_i} =d_i\log Y+O\Big(d_i\log\frac2{d_i}+\tau_i\Big),$$

with an absolute implied constant and the entropy term read as $0$ when $d_i=0$, and such that $\sum_{n_i\le X}\tau_i/n_i=o(\log X)$. The note proves neither the hypothesis nor a replacement for it: its Proposition 6.3 shows that prime-power towers of moduli produce harmonic spikes far above the entropy scale, so the charges cannot be omitted, and its abstract says that it isolates a missing lemma rather than a complete proof. The hypothesis is unproven, so this page derives nothing for the problem's standing.

Standing. Przemek Chojecki posted the note in the site's thread on 19 March 2026 as the outcome of 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). The claim stays claimed. The note's two unconditional special cases are the partial claim on its own page.

Depends on. Nothing on the wiki.