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Claim. For $k\ge1$ let $D_k$ be the set of integers that occur as a denominator in some representation $1=1/n_1+\cdots+1/n_k$ with $1\le n_1<\cdots<n_k$, and let $v(k)$ be the least integer $m>1$ not in $D_k$, as in the problem page's corrected Statement. Van Doorn and Tang's Theorem 1.1 states that there is an absolute constant $c>0$ with

$$v(k)\ \ge\ e^{ck^2}$$

for every positive integer $k$; the proof gives $c=\min{\log2/432^2,
1/(433C)^2}$ with $C$ the constant of Vose's theorem. The argument proves the nesting $D_k\subseteq D_{k+1}$ for $k\ge2$ (their Lemma 2.1, by splitting the largest denominator with $1/n=1/(n+1)+1/(n(n+1))$ and a variant for composite entries) and then applies Vose's theorem that every $a/b\in(0,1)$ is a sum of at most $C\sqrt{\log b}$ distinct unit fractions with denominators of a special form. The authors write that theirs is the first lower bound in the literature and that extracting the monograph's $v(k)\gg k!$ from the Bleicher–Erdős papers does not seem straightforward to them. On the upper side, their inequality (1.2), $v(k)\le|D_k|+2\le kF(k)+2$ with $F(k)$ the number of $k$-term representations, combined with the Elsholtz–Planitzer bound on $F(k)$, gives $v(k)\le c_0^{(2/5+o(1))2^k}$ with $c_0=1.26408\ldots$ the Vardi constant (the paper prints the exponent $(1/5+o(1))2^k$, pairing the exponent of Elsholtz and Planitzer's Corollary 3(2), stated for $c_0^2=1.5979\ldots$, with the Vardi constant; see the normalization remark on Problem 148). The statements are recorded on the source card; the proof of Theorem 1.1 is recorded there as a sketch and is not verified in this corpus.

Covers. The bounds $e^{ck^2}\le v(k)$ for every $k\ge1$ and $v(k)\le c_0^{(2/5+o(1))2^k}$ for large $k$. The lower bound is the best one valid for every $k\ge1$; for all large $k$ it is exceeded by the OpenAI release's $e^{e^{k/600}}$, recorded on its claim page, whose threshold is existential. The upper bound is the best recorded. Not settled: the growth of $v(k)$ beyond these bounds; the paper's Section 3 expresses the expectation, not a theorem, that the conjecture of Problem 304 would give $v(k)\ge e^{e^{ck}}$.

Depends on. No page of this wiki.

Acceptance. The paper is published in Mathematical Proceedings of the Cambridge Philosophical Society (online 8 July 2026, pp. 1--9, DOI 10.1017/S0305004126102102; the arXiv record's journal reference and the Crossref record agree), which is the refereed evidence; arXiv v2 (24 May 2026; v1 26 December 2025, the date of this page) records acceptance with minor revisions in its comments line, and the published text has not been compared with it. The site's commentary credits van Doorn and Tang with the lower bound, but the site labels the problem OPEN, so that commentary is not acceptance and reviewed is not listed. The second author announced the result in the problem's discussion thread on 29 December 2025.