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Claim. Three of the six questions of Problem 1209 are answered no. Questions (i) and (ii): for every function $g:\mathbb N\to\mathbb N$ there is a strictly increasing sequence of primes $b_1<b_2<\cdots$ with $b_k>g(k)$ for all $k$ such that $n=0$ is the only integer $n$ for which every $n+b_k$ is prime, and also the only integer for which every $n+b_k$ is squarefree (the note's Theorem 2.1); translating the sequence moves the unique shift to any prescribed positive integer (Corollary 2.2). So one shift making every term prime, or every term squarefree, does not force a second one, however fast the sequence grows. Question (iii.a): no integer $n$ makes $n+2^{2^k}$ prime for every $k\ge0$ (Theorem 3.1). For even $n$ the terms are even and eventually exceed $2$; for $n=1$ they are the Fermat numbers and $641\mid2^{32}+1$ (Lemma 3.2); for odd $n\ge3$, if $p=n+2^{2^t}$ is prime with $t\ge v_2(n-1)$, then $2^{2^t}$ has odd multiplicative order $M$ modulo $p$, so an $L$ with $2^L\equiv1\pmod M$ gives $p\mid n+2^{2^{t+L}}$, a larger multiple of $p$ (Lemma 3.3). The source is E. Barschkis, Erdős Problem #1209, a six-page note dated 15 April 2026 in the repository linked above, at its head of 15 April 2026, with the elementary checks the problem page records. The author writes in the thread that the ideas were explored with GPT Pro, the site's commentary credits the (iii.a) argument to the author and GPT, and the formal-conjectures docstring says the Lean file was written using ChatGPT; the claimant is the author, who published the note.

Covers. Questions (i), (ii) and (iii.a), each answered no, for every integer shift (the site's own construction, the curator's pending partial claim on its claim page, covers every integer shift for primes and nonnegative shifts for squarefree values). Not covered: questions (iii.b), (iii.c) and (iii.d), whether some $n$ makes $n+2^{2^k}$ always squarefree, infinitely often prime, or infinitely often squarefree; these stay open, and at $n=1$ the first two are the squarefreeness of every Fermat number and the infinitude of Fermat primes.

The site's construction. The commentary gives its own counterexample to (i) and (ii), on the page since its edit of 8 April 2026 by the site's revision history, a variant of the construction of Problem 429: $a_1=2$ and, for $k\ge2$, a prime $a_k>a_{k-1}$ with $q_k\mid a_k+k$ for a prime $q_k\nmid k$, so that the shift $k\ge2$ gives a composite term and the shift $1$ an even one, leaving $0$ as the only nonnegative shift for primes; $q_k^2$ in place of $q_k$ gives a non-squarefree term at each shift $k\ge2$, leaving at most the shifts $0$ and $1$ for squarefree terms. For primes, the choice $a_1=2$ already excludes every negative shift. The note's Theorem 2.1 adds the negative shifts for squarefree values and makes $0$ the unique shift for both properties with one sequence. Under the convention that shifts are positive, the site's prime sequence has no good shift at all, and its squarefree sequence at most the uncontrolled shift $1$. So both must be translated, as the note's Corollary 2.2 does, before they answer the questions. For (iii.a) the commentary records the order argument with the choice of $k$ large, which is the note's condition $t\ge v_2(n-1)$.

Standing. Pending. The site's commentary states (iii.a) proved and credits the note's author and GPT with the proof, in a paragraph that, by the site's revision history, entered the page in the edit of 17 April 2026, in response to the comment of 15 April 2026; the same history shows the construction answering (i) and (ii) on the page since the edit of 8 April 2026, before the note appeared. In the thread of 17 April 2026 the curator, T. F. Bloom, who did not write or submit the note, held that the construction in the remarks answers (i) and (ii) under the site's convention that shifts are positive integers, and that this construction can be trivially altered to allow negative shifts, which is the modification the note makes. The site's page-level label is OPEN with no per-part label, so this credit is commentary, not acceptance, and no reviewed evidence is listed. A forum comment of 16 April 2026 reports a routine check of the note that found no issues; it is noted, not counted. Not refereed: the problem page's search found no journal version. Not formalized in this corpus's sense: the note's Lean file at the pinned commit proves main_diagonal, corollary_unique_shift and no_universal_prime_shift with no sorry or axiom in its text and one native_decide, and the formal-conjectures statement file (linked from the problem page, not a formalization of this result) marks parts (i), (ii) and (iii.a) research solved, with a formal_proof attribute on (iii.a) only and,sorry bodies throughout (since 19 September 2026 the file proves parts (i) and (ii) itself); the files were neither built nor audited for statement fidelity here. The same order argument is the official 2015 solution of ELMO Problem 4, which settles (iii.a) for every shift $n\ge2$ (its claim page); the note also treats the shifts $n\le1$. Nothing here is this project's own review.

Depends on. No page of this wiki.