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Claim. Let $\gamma>1$ be the Salem number with minimal polynomial $x^{18}-x^{12}-x^{11}-x^{10}-x^9-x^8-x^7-x^6+1$, so that $6/5<\gamma<13/10<(1+\sqrt5)/2$. There are arbitrarily large real $t>0$ such that $\lfloor t\gamma^n\rfloor$ is even for every $n\ge0$; such a sequence represents no odd integer and is not complete. This is Theorem 9 of Geneson, J., Deletion thresholds and exponential examples for complete sequences, arXiv:2609.25107 (v1 2026-09-20), recorded on the card Theorem 9, whose one-page proof has not been independently reviewed; the ResearchGate note of the forum claim, "Incomplete sequences from powers of a Salem number", is the earlier posting of the same result, whose posting date is not recorded, so the page is dated by the forum claim. Since every term is even, the sequence is not complete under the corrected Statement of the problem page, nor under the site's wording, which counts equal values once.

Submission note. Posted to erdosproblems.com as a proof claim by Jesse Geneson (account jtg) on 6 September 2026, giving "GPT-6 Astra, Claude Opus 5" as the AI used:

I have posted a short note applying a theorem of Dubickas (2006) to disprove the assertion that $(\lfloor t\gamma^n\rfloor)_{n\ge1}$ is complete for every $t>0$ and $1<\gamma<(1+\sqrt5)/2$. With an elementary sign adjustment, the theorem gives a Salem number in this range and positive $t$ for which every term is even. Notes: The proof in this paper was found by Codex with GPT-6 Astra Ultra. The author designed and directed a workflow with two teams of Codex agents, one responsible for writing and revising and the other for auditing the proof. The teams worked through successive rounds of revision and review until the referee agents accepted the manuscript under the author’s guidelines for correctness, completeness, clarity, and exposition. The author then ran a separate iterative writer-referee process, with a team of Codex agents using GPT-6 Astra Ultra as writers and Claude Opus 5 as referee, continuing until the referee accepted the revised manuscript. The author subsequently read the proof, edited it, and requested further revisions for clarity. The author takes full responsibility for the content of the article.

Covers. The conjecture, of Graham (1964) and of Erdős and Graham's 1980 monograph and repeated in the site's remarks, that $\lfloor t\alpha^n\rfloor$ is complete for every $t>0$ and every $1<\alpha<(1+\sqrt5)/2$: it is false at one base $\gamma\approx1.25$ for unspecified large coefficients. The claim does not classify the pairs $(t,\alpha)$ that Problem 349 asks for, says nothing at other bases, and gives no explicit coefficient; the paper notes that van Doorn's computer-assisted completeness region (the row of van Doorn's Proposition 8 giving completeness for all $t\le5$ when $1.2<\alpha\le1.3$, recorded on van Doorn's claim page and on the Theorem 9 card) forces every such coefficient at this base past $5$. The claim's value is disproved: the result refutes a conjectured completeness region and proves no completeness.

Argument. As the paper describes it: a theorem of Dubickas (2006) gives, for a Pisot or Salem number whose minimal polynomial $P$ has $P(1)=-q$ with $q\ge2$, an element $\xi$ of $\mathbb Q(\gamma)$ whose multiples $\xi\gamma^n$ all have fractional part within any prescribed distance of $1/q$; an elementary sign adjustment (Proposition 11) makes the multiplier positive when $q\ge3$, and doubling it makes every floor even. Here $P(1)=-5$, and two sign checks place the root in $(6/5,13/10)$.

Standing. Claimed. A preprint with no journal record on the arXiv page; the site's label is OPEN as of 2026-10-06, and no outside review is known. The author submitted the claim to the thread on 2026-09-06 (the discussion link), disclosing that the proof was found by Codex with GPT-6 Astra Ultra under a writer-and-referee workflow the author directed, with Claude Opus 5 as a referee, and that the author read and edited the proof and takes responsibility for it. A forum comment of 2026-09-14 reported explicit values obtained with ChatGPT, a coefficient $t\approx6.5\cdot10^9$ at this base and a second pair at the Salem number $\approx1.1883$, and asked whether a smallest such base exists; the values are a thread post. No review, refereed publication or Lean proof of the result exists, so the claim lists no evidence.

Depends on. Nothing in this wiki.