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Claim. In a post of 10 October 2026 on the erdosproblems.com forum, Wouter van Doorn announced his write-up Short Egyptian fractions, recorded on Problem 304's claim page, and stated that the write-up's proof "immediately gives $v(k) > e^{e^{ck}}$" for the smallest missing denominator $v(k)$ of Problem 293, "which in turn implies $F(k) > e^{e^{ck}}$" for Problem 148, both "optimal up to the value of $c$". The post gives no value of $c$ and no range of $k$. The step to $v(k)$, a greedy start that never takes $1/b$ applied to $(b-1)/b$, is a sketch, recorded with what it leaves out on Problem 293's claim page. The step from $v(k)$ to $F(k)$ is van Doorn and Tang's inequality (1.2), $v(k)\le kF(k)+2$, which turns $v(k)>e^{e^{ck}}$ into $F(k)>e^{e^{c'k}}$ for every $c'<c$ and all large $k$; the post does not name it.

Submission note. Posted to the erdosproblems.com forum by Wouter van Doorn (account Woett) on 10 October 2026, in the comment thread of the curator's blog post "Erdős problems in the OpenAI proofs":

Human-generated exposition

With the help of ChatGPT I have substantially simplified OpenAI's solution to #304. You can find my write-up here, proving that a fraction $\frac{a}{b} \in (0, 1)$ can be written as the sum of at most $\left(\frac{16}{\log 2} + o(1) \right) \log \log b$ distinct unit fractions.

Optimality

The constant of $\frac{16}{\log 2} \approx 23.08$ is pretty good (compare e.g. with the constant $\frac{257}{\log 2} \approx 370.77$ that appears in the OpenAI paper), but not optimal. Indeed, with more effort (and by relying on various results from the literature on the distribution of prime numbers) one can lower it to $\frac{7}{\log 2} \approx 10.10$, and perhaps further still. I tried to keep the write-up as simple as possible, however, so I decided that $\frac{16}{\log 2}$ is good enough for now.

Applications

In the proof we start with a greedy expansion for the first $\left(\frac{1}{\log 2} + o(1) \right) \log \log b$ terms. Now, it is trivial to do this greedy start but with taking $\frac{1}{b+1}$ instead of $\frac{1}{b}$ if the latter would be correct according to the greedy strategy. With this slight modification, if we started with the fraction $\frac{b-1}{b}$, then we can simply add $\frac{1}{b}$ at the end to obtain a decomposition of $1$ as a sum of distinct unit fractions that contains $\frac{1}{b}$, which is what we aim for in #293. Long story short, in the notation of #293, the linked proof immediately gives $v(k) > e^{e^{ck}}$, which in turn implies $F(k) > e^{e^{ck}}$ for #148 as well, both of which are optimal up to the value of $c$. Three solutions for the price of one!

Some words on the OpenAI paper

It's.. Bizarre. In the first 5 sections of the paper they prove that $\ll \log \log b$ unit fractions suffice. The proof takes about 20 pages and seems quite complicated; I haven't even tried to understand any of it. Section 6 and 7 then apply the proven results on #304 to #148 and #293 respectively. So far so sensible (despite the apparent difficulty).

But then, Section 8 aims to resolve #293 with the explicit constant of $\frac{257}{\log 2}$. This is done by a completely different and conceptually much easier argument! For some reason #304 and #148 are not even mentioned in this section, even though this proof works just as fine for either. And seemingly, the formalisations of the solutions for the latter two problems go through the original much more difficult proof. The formalisation of #293 on the other hand, does use this easier proof. The only problem is, this easier proof relies on Vinogradov's theorem on the odd Goldbach conjecture, which is probably hard to formalise. So the formalisation actually modifies the easy proof in a way as to avoid Vinogradov's theorem.

In any case, the write-up I linked gives (a simplification of) the relatively easy Vinogradov-reliant proof, with the first 7 sections of the OpenAI paper completely irrelevant and ignored.

Covers. A doubly exponential lower bound $F(k)>e^{e^{ck}}$ for some $c>0$, the form of the lower half of the release's accepted Corollary 1.2, $\log\log F(k)\ge ck$ for all large $k$. With no value of $c$, the claim adds nothing to that bound. It says nothing about the upper bound, and the problem's request for good estimates stays open.

Depends on. The same post's claim on Problem 293, from which the bound on $F(k)$ is deduced, and through it van Doorn's write-up on Problem 304, which mentions neither problem.

Standing. Rejected: the claim does not improve the best known bound. Lower bounds of this form already rest on the release's accepted Corollary 1.2, and with no value of $c$ the claim adds nothing to it. The claim is also a sketch in a forum post: the deduction of the bound on $v(k)$ is not written out in the write-up or elsewhere, and the corpus has not checked it. No review exists. The curator replied to the post on 10 October 2026 with thanks, which is commentary, not acceptance. Bloom's catalog labeled the problem OPEN (page last edited 27 September 2025).