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Claim. Problem 951, read with the inequality $\#\{a_i\le x\}\le\pi(x)$ required for every $x\ge1$, fails at $x$ just below each of the primes $5$, $7$, $11$, $13$ and $17$. The note of 28 January 2026 gives three rational generators $101/42\approx2.40$, $367/103\approx3.56$ and $113/24\approx4.71$ whose power products are pairwise at least $1$ apart, found by computational Diophantine approximation (Matveev's lower bound for linear forms in logarithms and LLL reduction, in the manner of de Weger); since $113/24<5=p_3$, three generators lie below $x$ for $x\in[113/24,5)$, where $\pi(x)=2$, and the extension step of the 27 January 2026 note makes them the start of an infinite sequence with the separation property. The note of 1 February 2026 gives, for each $3\le n\le7$, $n$ generators below the $n$-th prime $p_n$ with the separation property, by a probabilistic construction certified by interval arithmetic along the lines a thread comment of 28 January 2026 proposed, so the inequality fails at $x=5-\varepsilon$, $7-\varepsilon$, $11-\varepsilon$, $13-\varepsilon$ and $17-\varepsilon$. Both notes were written with ChatGPT, the second in what the author describes as a collaboration of forty-four turns; they are documents on a file-sharing service, linked above, and this page records the claim as the thread describes it, not from the documents.

Covers. The part every_x of the problem page, the reading in which the inequality must hold for every $x$, at the values of $x$ named above. That part is also refuted at $x=10$ by the pending claim, so this claim adds instances and changes no standing. It says nothing about the part large_x, the reading for all sufficiently large $x$; the notes' authors expect failures at arbitrarily large $x$ but have no proof.

Depends on. [[problems/number_theory/E0951/claims/2026_01_27_leeham|Leeham's counterexample]], whose greedy extension step turns each finite set of generators into an infinite sequence with the separation property.

Standing. Claimed. The site's commentary (page last edited 06 April 2026) credits the $x=10$ counterexample only and does not mention these notes; they are not refereed, not registered on the site's proof-claim tab and not reviewed by anyone named. No independent check of their numerical certificates is recorded; the three rational generators of 28 January 2026 can be checked by exact enumeration of their products up to any bound.