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Claim. Question (i) of Problem 1209 is answered no for every integer shift, and question (ii) for nonnegative shifts, by a variant of the construction of Problem 429. Take $a_1=2$ and, for $k\ge2$, a prime $a_k>a_{k-1}$ with $a_k+k\equiv0\pmod{q_k}$ for some prime $q_k\nmid k$; Dirichlet's theorem supplies such primes, since $\gcd(k,q_k)=1$, and $a_k$ may be taken as large as any prescribed growth demands. For $n=0$ every $n+a_k$ is prime. For $n=k\ge2$, $k+a_k$ is a multiple of $q_k$ and exceeds $q_k$ once $a_k$ is chosen above $q_k$, so it is composite; for $n=1$, $1+a_2$ is even and greater than $2$. Every negative shift fails at $k=1$, since $n+a_1=n+2\le1$. So $0$ is the only integer shift making every term prime, and one such shift does not force infinitely many, however fast the sequence grows. With $q_k^2$ in place of $q_k$ the term $k+a_k$ is divisible by $q_k^2$ for every $k\ge2$, so the nonnegative shifts making every term squarefree are among $0$ and $1$, finitely many, which answers (ii). The commentary states the construction and its growth in two sentences; the checks of the shifts $n=1$ and $n=k$ above are the problem page's elementary reading of it.

Covers. Question (i) for every integer shift, since a negative shift fails at $k=1$ ($n+a_1=n+2\le1$), and question (ii) for nonnegative shifts: one shift making every $n+a_k$ prime, or one nonnegative shift making every $n+a_k$ squarefree, does not force infinitely many. Under the convention that shifts are positive, the sequences must first be translated (Barschkis's Corollary 2.2). Not covered: negative shifts in (ii), which the construction does not exclude ($n=-1$ gives $n+a_1=1$, and $a_k-1$ is uncontrolled) and Barschkis's note does, on its claim page; question (iii.a), which the commentary credits to Barschkis and GPT; and questions (iii.b), (iii.c) and (iii.d), which stay open.

Claimant and date. The construction is the commentary of the site's own problem page, unsigned, whose recommended citation names the site's curator, T. F. Bloom, as the page's author; the page is filed under that name. The site's revision history shows two superseded revisions: the construction entered the page in the edit of 8 April 2026 (the revision without it was superseded at 14:19 UTC that day), a week before Barschkis's note of 15 April 2026, and the paragraph on (iii.a) entered in the edit of 17 April 2026 (07:27 UTC). In the thread of 17 April 2026 the curator wrote that the first two questions were already answered by the construction in the remarks.

Standing. Pending. The label is the claimant's own: the curator who labels the problem wrote the construction, so the page's label is no independent review and no reviewed evidence is listed. No paper or preprint of the construction exists (the proof-claim tab was empty on 2026-09-18). Since 19 September 2026 (formal-conjectures pull request 6065), the collection's own file proves its statements erdos_1209.parts.i and .parts.ii, over shifts in $\mathbb N$. It uses a variant of this construction: Dirichlet primes $p\equiv-k\pmod q$ for a prime $q>k$, so that the shift $k$ fails at the $k$-th term. The pull request calls it the construction in the docstring. This corpus has not built or audited that Lean, so it is a formalization link, not formalized evidence. Barschkis's note also excludes every nonzero integer shift for squarefree values, and its Lean file proves that statement; its standing is recorded on its own page and is not evidence for this one. Nothing here is this project's own review.

Depends on. No page of this wiki.