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Claim. For integers $1\le a<b$ let $N(a,b)$ be the least number of distinct unit fractions with denominators above $1$ whose sum is $a/b$, and $N(b)=\max_{1\le a<b}N(a,b)$, the extremal function of Problem 304. Then

$$N(b)\le\Bigl(\frac{16}{\log2}+o(1)\Bigr)\log\log b\qquad(b\to\infty),$$

so $N(b)\ll\log\log b$ and the problem's question is answered yes. With Erdős's lower bound of 1950, recorded on its claim page, $N(b)\asymp\log\log b$. This is Theorem 2 (p. 2) of Wouter van Doorn, Short Egyptian fractions, a six-page write-up added on 9 October 2026 (UTC, the claim's date) to his GitHub repository Woett/Mathematical-shorts at the commit linked above. The theorem states that every fraction $a/b\in(0,1)$ is a sum of at most that many distinct unit fractions. The bound is uniform in $a$, and no denominator is $1$ since $a/b<1$. The abstract calls the write-up a human-written exposition and simplification of a proof recently posted by OpenAI, the release manuscript recorded on its claim page.

The proof has three steps. First, greedy steps keep the denominator unreduced, $b_i=bx_1\cdots x_i$, and stop when the remainder vanishes or $b_r\ge n$. Here $n=\operatorname{lcm}(1,\ldots,t)^2$ with $t$ about $\log b(\log\log b)^2/(2c)$. The remainder after $i$ steps is at most $2^{-2^{i-1}}$, so this uses at most $(1/\log2+o(1))\log\log b$ unit fractions (p. 2). Second, Lemma 3 (p. 3) says that $b_r$ has a divisor in $[\mu,b\mu]$ for every real $1\le\mu\le b_r$. Repeated division by such divisors writes $a_rn=d_1g_1+\cdots+d_sg_s$ with $d_j\mid b_r$, $1\le g_j\le B=\exp(c\log n/\log\log n)$ and $s\le(1/c+o(1))\log\log b$ (pp. 3--4). Third, Lemma 1 (p. 2, proved on pp. 5--6) says that for $0<c<\tfrac23\log2$ and large $t$, every positive integer up to $B$ is a sum of at most $10$ fractions $e/f$ with $e\mid n$. So each $d_jg_j/(b_rn)$ splits into at most $10$ unit fractions, and the total is at most $(1/\log2+10/c+o(1))\log\log b$. This tends to $16/\log2$ as $c$ tends to $\tfrac23\log2$. By Takenouchi's theorem of 1921, as the write-up states it (p. 4), the unit fractions can then be made distinct with the same number of terms. The proof of Lemma 1 writes each large odd $m\le B$ as a sum of three primes by Vinogradov's theorem. The three primes avoid the few primes $p$ modulo which the divisors of $P$, the product of the first $2+\lceil\frac34\log_2m\rceil$ primes, occupy fewer than $p^{3/4}$ nonzero classes. Modulo each of the three primes, Lemma 4 (pp. 4--5, a Cauchy–Schwarz count) writes $-1$ as $u_1v_1+u_2v_2$ with $u_i,v_i\mid P$. So each prime is a sum of three fractions whose numerators divide $P^2$, which divides $n$. An even $m$ adds the fraction $1/1$.

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.

Depends on. Nothing in this wiki. The write-up proves its lemmas itself. Its outside inputs are the prime number theorem, Vinogradov's three-prime theorem (cited through Kumchev and Tolev's survey) and Takenouchi's 1921 theorem on making a sum of unit fractions distinct. The write-up asserts without derivation that every large odd $m$ has at least $m/(20\log^2m)$ pairwise disjoint representations as a sum of three primes. The proof needs only one representation that avoids the exceptional primes, and the release's proof of its Lemma 8.2 obtains one by counting ordered triples. The write-up does not use the release's theorems.

Relation to other claims. Section 8 of the release's manuscript proves, for Problem 293, that every large $m$ is a denominator in a representation of $1$ by at most $(257/\log2+\varepsilon)\log\log m$ distinct unit fractions (Proposition 8.1). Its two main lemmas are a supply of fractions whose numerators divide a fixed integer, resting on the three-prime theorem (Lemma 8.2), and the divisor property of the unreduced greedy denominator (Lemma 8.3), as the release's source card records. The write-up's Lemmas 1 and 3 play the same roles. Its Lemma 1 writes each prime with three fractions where Lemma 8.2 uses five, so it needs at most ten summands where Lemma 8.2 needs sixteen. It applies the construction to $a/b$ directly and does not use Sections 2--7 of the manuscript. The manuscript's Theorem 1.1, the accepted claim, leaves its constant unspecified. This is a different result from van Doorn and GPT-6 Astra Pro's note of 16 September 2026, whose bound $2c_0(\log\log b)^2$ came from practical numbers; this bound implies it.

AI use. The write-up declares no AI assistance for itself and calls itself a human-written exposition. The only AI system it names is ChatGPT, credited in Section 1 (p. 1) with the earlier practical-number approach of its reference [2], the note of van Doorn and GPT-6 Astra Pro. The forum post announcing it, under the heading "Human-generated exposition", says that the simplification was made with the help of ChatGPT. ChatGPT is credited only with assistance, so it is not listed as an author.

Not proved in the write-up. Two statements of the forum post are not in the write-up. The first is that the constant can be lowered to $7/\log2$ with more effort, using results on the distribution of primes; no argument is given. The second is that a greedy start that never takes $1/b$, applied to $(b-1)/b$, gives $v(k)>e^{e^{ck}}$ for Problem 293 and hence $F(k)>e^{e^{ck}}$ for Problem 148. The write-up mentions neither problem, so these consequences are sketches in the post. They are recorded as rejected partial claims on Problem 293's claim page and Problem 148's claim page. Lower bounds of this form are already accepted there, from the release's Corollaries 1.3 and 1.2, and with no value of $c$ the claims do not improve them.

Standing. Claimed. The write-up is a PDF on GitHub, not on arXiv and not refereed, with no Lean formalization, and no outside review is known. Here the proofs of Theorem 2 and Lemmas 1, 3 and 4 were read step by step against the problem's statement, and no gap was found. That reading takes as given the prime number theorem, Takenouchi's theorem and the asserted count of disjoint three-prime representations. It is the corpus's own reading, not an independent review, and it supplies no evidence. 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 29 December 2025). The problem's standing rests on the release's accepted theorem; if accepted, this claim would give a second proof, with an explicit constant.