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The claim. For all sufficiently large primes $p$,

$$f(p)\ \ge\ \log_2p+\Bigl(\frac1{2\log2}+o(1)\Bigr)\log\log p ,$$

where $f(N)$ is the least $k$ such that a uniformly random $k$-element subset of an abelian group of order $N$ covers the group by subset sums with probability at least $1/2$. The bound $f(N)\le\log_2N+o(\log\log N)$ asked for in Problem 543 therefore fails along the primes, with an explicit second-order term. The source is J. Ma and Q. Tang, An Erdős problem on random subset sums in finite abelian groups, arXiv:2602.05768 (v1 of 5 February 2026, v2 of 19 March 2026, 13 pages, described by its authors as the submitted version); Tang announced the paper and its bound on the site's thread for the problem on 6 February 2026; the abstract states the theorem, and the paper was not read here beyond it. Against it stands the upper bound of Erdős and Rényi (Erdős and Rényi 1965, Theorem 2), which the abstract gives as $f(N)\le\log_2N+(\tfrac1{\log2}+o(1))\log\log N$ for every $N$; if the claim holds, the second-order term for primes lies between $\tfrac1{2\log2}$ and $\tfrac1{\log2}$ times $\log\log p$.

Relation to the accepted claim. The qualitative negative answer, that a random subset of size $\log_2p+o(\log\log p)$ misses some element of $\mathbb F_p$ with probability tending to one, is the accepted claim Tang 2026, a note by one of this paper's authors written with ChatGPT; the repository holding that note says this paper supersedes it. The paper is a different author set's publication and asserts a stronger result, so it is recorded on its own page, and the acceptance evidence of the note does not extend to it.

Acceptance. None listed. The site's curator credits the disproof to ChatGPT and Tang on the problem page, last edited 27 January 2026, before this paper appeared, and names neither the paper nor its bound; the paper is a submitted version with no journal publication recorded; no Lean formalization is recorded; and this corpus has not verified the proof.

Depends on. No page of this wiki; the result rests on the cited paper.