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Claim. The answer to Problem 728 is yes: for every $0<C_1<C_2$ and every fixed $0<\varepsilon<1/2$ there are infinitely many triples of positive integers $(a,b,n)$ with $\varepsilon n\le a,b\le(1-\varepsilon)n$, $a!\,b!\mid n!\,(a+b-n)!$ and $C_1\log n<a+b-n<C_2\log n$, the same statement as on the accepted claim page Barreto 2026. Two manuscripts posted on 2026-07-13 to the site's discussion thread by Jeff Pickhardt (forum user JPickhardt), the claimant here, whose title pages name Pickhardt and the Paratelligent Research Agent as authors, say they prove it by arguments different from the probabilistic carry-counting of the January proofs. The first, A deterministic resolution of Erdős Problem #728 via small-prime annihilation, puts $N=a+b$ in a residue class $N\equiv-1$ modulo a product $M$ of small prime powers, so that by Lucas's theorem the row $\binom{N}{\cdot}$ of Pascal's triangle avoids every prime up to a constant multiple of $k=a+b-n$, and then counts the choices of $a$ that survive the medium and large primes. The second, An AI-derived proof of Erdős Problem #728 via higher-power carry compensation, rests on an exact small-prime congruence and on offsetting a missing carry at a prime $p$ by a surplus carry at a higher power of $p$. Both say the proofs were produced autonomously by the Paratelligent Research Agent, an AI system of Pickhardt's; the posting says its first paper, the higher-power-carry manuscript, was checked in Lean, and that manuscript gives its Lean repository, github.com/pickhardt/erdos-728, linked above at its commit of 2026-07-13, whose README declares it a formalization of that manuscript and reports that its main theorem erdos728Main_proved builds without sorry on propext, Classical.choice and Quot.sound only; this corpus has not built or audited it, so no formalized evidence is listed. On 2026-10-07 the manuscript links above redirected to pages on the agent's maker's site, paratelligent.com.

Depends on. No page of this wiki.

Standing. The manuscripts are hosted on the agent's own site and are not on arXiv or in a journal; no reviewer has acknowledged them, the site's curator credits the January proof, and the thread records no response. The claim therefore stays claimed; it adds nothing to the problem's standing, which the accepted pages carry.