Problem 304

../

claims/: The 5 claim pages of Problem 304, one per claimant's result; the problem's standing derives from them.

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Statement. For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with

$$\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$$

Estimate $N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that $N(b) \ll \log\log b$?

Formulation. $N(a,b)$ is Nakayama's notation, used by Erdős in 1950 (p. 193); it depends only on the rational $a/b$, and $N(a,b)\le a$ always (p. 193, display (4)). The maximum runs over all $1\le a<b$ with no coprimality condition; the formal statement below does the same. The average $\frac1b\sum_{1\le a<b}N(a,b)\gg\log\log b$ quoted by the site is a separate statement of the same 1950 paper.

Status. Open on the site: the label is OPEN (page last edited 29 December 2025; no proof claim on its tab). The standing derived from the claim pages is solved, claim proved: Theorem 1.1 of the OpenAI mathematics release's manuscript of 25 September 2026 gives $N(b)\le c_2\log\log b$ for every $b\ge b_0$, so $N(b)\ll\log\log b$ and, with Erdős's lower bound of 1950, $N(b)\asymp\log\log b$. The claim is accepted on formalized evidence alone, the Lean declarations the corpus's verification built and audited, as its claim page records; it has no outside review and no refereed publication. The earlier bounds have accepted partial claim pages on their refereed papers: Erdős's 1950 bounds $\log\log b\ll N(b)\ll\log b/\log\log b$, and Vose's 1985 bound $N(b)\ll\sqrt{\log b}$, which replaced Erdős's upper bound. Two pending partial claims do not change the standing: van Doorn and GPT-6 Astra Pro's squared double-logarithm bound of 16 September 2026, implied by the accepted bound, and a Lean proof by Harmonic's Aristotle prover of the lower bound $\log\log b\le6N(b)$, published in 2026 and not built here. The conjecture $N(b)\ll\log\log b$ was stated by Erdős in 1950 and repeated in the 1980 monograph.

Source. erdosproblems.com/304, accessed 2026-09-17: the label OPEN (page last edited 29 December 2025), one discussion comment and no proof claim. Cite as: T. F. Bloom, Erdős Problem #304, https://www.erdosproblems.com/304, accessed 2026-09-17.

References.

• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), pp. 37--38.
• [Er50c] Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210; Theorems 1 and 2, p. 195; proofs pp. 198--203 and 208--209; English summary p. 210.
• [Vo85] Vose, M. D., Egyptian fractions. Bull. London Math. Soc. 17 (1985), no. 1, 21--24, doi:10.1112/blms/17.1.21; Zbl 0558.10015. Proves that every $a/b<1$ is a sum of $\ll\sqrt{\log b}$ distinct unit fractions.
• [vDTa25b] van Doorn, W. and Tang, Q., The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length. arXiv:2512.22083 (v2 24 May 2026); Math. Proc. Cambridge Philos. Soc., published online 8 July 2026, doi:10.1017/S0305004126102102; Section 3.
• [Yo92] Yokota, H., On a sum of divisors. Canad. Math. Bull. 35 (1992), no. 3, 423--430, doi:10.4153/CMB-1992-056-7. Linked from the site's discussion.
• OEIS A097847 and A097849, linked by the site.

Formalization. Statement only. The file ErdosProblems/304.lean of formal-conjectures, at the linked commit of 18 September 2026, defines unitFractionExpressible a b as the set of sizes of finite sets of integers above $1$ whose reciprocals sum to $a/b$, smallestCollection a b as its infimum and smallestCollectionTo b as the supremum over a ∈ Finset.Ico 1 b, and declares upper_bound : answer(sorry) ↔ (fun b : ℕ => (smallestCollectionTo b : ℝ)) =O[atTop] (fun b : ℕ => Real.log (Real.log b)) under category research open, with the Erdős 1950 bounds and the Vose bound as research solved variants whose proofs are sorry. Since that commit the lower-bound variant erdos_304.variants.lower_1950 carries a formal_proof annotation pointing to the Lean proof by Harmonic's Aristotle prover recorded on its claim page, while upper_bound stays research open. The definitions match the site's $N(a,b)$ and $N(b)$. The site's note that the problem is formalized in Lean refers to this statement file; it is not a proof, and neither it nor the annotated proof is built or audited here. On 2026-09-17 the community database recorded a formalized statement and no formal proof. The release's formalization of Theorem 1.1, built and audited here, is recorded on the claim page linked under Status.

Current assessment

The question. The site states the problem as above, shows OPEN, cites [ErGr80, p. 37], attributes $\log\log b\ll N(b)\ll\log b/\log\log b$ to [Er50c] and $N(b)\ll\sqrt{\log b}$ to [Vo85], records the average bound, relates the problem to Problem 18 and to Problem 293 through [vDTa25b], and notes the Lean statement. The monograph's pp. 37–38 (Erdős–Graham 1980) state that Erdős proved $c\log\log b<N(b)<c'\log b/\log\log b$, improving de Bruijn's unpublished $c'\log b/\log\log\log b$; that "The true order of $N(b)$ seems very hard to determine. Even showing that $N(b)=o\left(\frac{\log b}{\log\log b}\right)$ would be of interest"; that the upper bound follows from the lemma that every number less than $n!$ is a sum of fewer than $n$ distinct divisors of $n!$, and that "No doubt very many fewer than $n$ divisors are required when $n$ is large, perhaps even only $(\log n)^c$, which would then imply $N(b)<c'\log\log b$"; and, on p. 38, that $n(b)=\frac1b\sum_{a=1}^bN(a,b)$ satisfies $n(b)>c\log\log b$.

Claims. Five claim pages. The first, the OpenAI release's Theorem 1.1 (25 September 2026), status accepted, scope full, claim proved: $c_1\log\log b\le N(b)\le c_2\log\log b$ for all $b\ge b_0$, with the constants not made explicit. The page names the two Lean declarations the corpus's verification built, their axioms and the comparator challenge that pins them; the manuscript is unrefereed, unreviewed outside the repository and attributed by the release to an internal model. The result replaces Vose's upper bound below, implies the claimed $(\log\log b)^2$ bound of the van Doorn note and the superseded $(\log b)^{\varepsilon}$ deduction below, and settles both the estimate and the question of the statement. The second, van Doorn and GPT-6 Astra Pro's Theorem 1.2 (16 September 2026), status claimed, scope partial, claim proved: $N(b)\le2c_0(\log\log b)^2$ for all large $b$, $c_0=14/\log2$, registered on Problem 18's proof-claims tab and implied by the first. The third and fourth are accepted partial claims on refereed papers: Erdős's 1950 bounds, Theorems 1 and 2 of [Er50c], $N(a,b)<c_1\log b/\log\log b$ for every $0<a<b$ and $N(b-1,b)>\log\log b-1$ with the average bound; and Vose's 1985 bound, $N(b)\ll\sqrt{\log b}$ [Vo85]. The fifth, a Lean proof of the lower bound by Harmonic's Aristotle prover, status claimed, scope partial, claim proved: $\log\log b\le6N(b)$ for $b\ge3$, published by the GitHub account thepriceisright and annotated since 18 September 2026 as the formal_proof of the formal-conjectures lower-bound variant; the file names no informal author, so it is an independent proof with its own page, and it is not built here. None of the partial claims changes the standing.

Origin. Erdős's 1950 paper (in Hungarian): Theorem 1 (p. 195) gives $N(a,b)<c_1\log b/\log\log b$ for all $0<a<b$, with $c_1=8$ for $b>4096$ (p. 202), by writing $a/b$ over $n!$ with $(n-1)!<b\le n!$ and using that every integer below $n!$ is a sum of at most $n$ distinct divisors of $n!$. On the same page Erdős writes that he considers it probable that this can be sharpened to $N(a,b)<c'\log\log b$, which is the conjecture of this problem (van Doorn and Tang, p. 6, say the same), and that no sharpening beyond $\log\log b$ is possible because of Theorem 2 (p. 195): $\frac1{b-2}\sum_{a=1}^{b-2}N(a,b)>\frac12(\log\log b-1)$ and $N(b-1,b)>\log\log b-1$ for every positive integer $b$. The English summary on p. 210 repeats all three statements.

Known results.

A claimed weaker bound, not adopted. A mostly AI-generated note carded as van Doorn 2026 claims, as its Theorem 1.2, that every fraction $a/b$ with $b$ large is a sum of at most $2c_0(\log\log b)^2$ distinct unit fractions, $c_0=14/\log 2$; its claim page is 2026_09_16_van_doorn. The note is unrefereed, its authors' Lean formalization is not built here, and the claim is not adopted into the bounds above; the accepted bound under Claims implies it.

Site and database state. The site shows OPEN (last edited 29 December 2025), one comment and no proof claim; the van Doorn claim above is posted on the proof-claims tab of Problem 18 (submitted 2026-09-16; as of 2026-10-05 no comments, not accepted, not on arXiv, and no commits to its repository after 16 September 2026); the community database says open; formal-conjectures marks upper_bound research open and, since 18 September 2026, annotates its lower-bound variant with the formal_proof recorded on the Aristotle proof's page; there is no Palomar entry; and on 2026-10-06 conjectures.io offered a live bounty, "Erdős problem 304 - upper bound", for the full $O(\log\log b)$ statement with no attempt recorded. None of these records a proof or disproof of $N(b)\ll\log\log b$; the release manuscript of 25 September 2026 is recorded under Claims.

• Lower bounds: Theorem 2 gives $N(b)\ge N(b-1,b)>\log\log b-1$ and the average bound (claim page); its proof shows that a representation of $1$ containing $1/b$ with $n$ terms forces $b<s_n$ for the Sylvester sequence $s_1=2$, $s_{j+1}=s_j(s_j-1)+1$ (Theorem 5 of the paper), so $n>\log\log b$. The Lean proof on the Aristotle proof's page reaches $\log\log b\le6N(b)$ for $b\ge3$ by a different route.
• Upper bounds: Theorem 1, and Vose's $N(b)\ll\sqrt{\log b}$ [Vo85] (claim page). As the review Zbl 0558.10015 describes it, Vose first shows, by an argument of Erdős's 1950 paper, that there is an increasing sequence $N_k$ such that every integer $1<m<N_k$ is a sum of at most $O(\sqrt{\log N_{k-1}})$ distinct divisors of $N_k$, and derives the bound from it. Van Doorn and Tang restate it as their Lemma 2.2 (every $a/b\in(0,1)$ is a sum of at most $C\sqrt{\log b}$ distinct unit fractions whose denominators divide, or are $b$ times divisors of, a number $N_K=4^{\alpha K^2}(p_1\cdots p_K)^2$) and as their display (3.1), and Liu and Sawhney restate it (arXiv:2404.07113v1, p. 3). The paper itself is not held.
• The link with Problem 293 (van Doorn–Tang, Section 3): if $b<v(k)$ then some $k$-term representation of $1$ contains $1/b$, so $N(b-1,b)\le k-1$; and if $N(b)\ll\log\log b$ held, the authors write that "it seems likely" their method would give $v(k)\ge e^{e^{ck}}$. The first is a two-line argument, the second an expectation; neither changes the status. Their Theorem 1.1 uses Vose's construction.
• The site's one comment (6 February 2026, user Alfaiz) suggests that a 1992 paper of Yokota on a sum of divisors is related and links its publisher PDF; its bearing on $N(b)$ is not established here, and it is recorded as a lead only.

Finite values (leads). OEIS A097847 (the triangle $T(n,k)$ of the least number of unit fractions for $k/n$, $1\le k\le n$) and A097849 (its row maxima, which equal $N(n)$ since the diagonal entry is $1$) listed, 105 row maxima, which never exceed $6$; the first $6$ occurs at $n=79$; the entries note that $T(17,4)=3$ is smaller than the greedy count and (a comment of May 2026) that the row maximum need not be attained at an $a$ coprime to $n$, the first case being $n=42$. A public report of 26 July 2026 by Patrick White with Claude (Anthropic), as the report names its authors (erdosproblemaday.com/report/304), claims $N(b)\le7$ for $b\le1000$ with equality at seven values, labels itself partial, and says the uniform question is open; the computation is not verified here.

Search scope. Routes; none found a proof or disproof of $N(b)\ll\log\log b$.

• The site's three pages; the community database (open; formalized statement; no proof URL); formal-conjectures 304.lean at the pinned commit.
• arXiv: the abstract page of 2512.22083; API metadata searches for "Egyptian fractions" AND "number of terms" (two records), "unit fractions" AND denominators AND distinct (eleven) and a sweep of 2025–2026 abstracts mentioning "Egyptian fractions" or "unit fractions" (29 records); none on $N(b)$.
• Crossref: the records of [Vo85] (Wiley, 1985; cited by ten works in its count) and [vDTa25b]. Semantic Scholar: Vose's DOI is not indexed.
• OEIS A097847 and A097849.
• The primary sources: [Er50c]; [vDTa25b] (arXiv v2); [ErGr80] pp. 37–38.

Not searched: MathSciNet, Google Scholar, X.

Proof coverage. Theorems 1 and 2 of 1950 are paged with locators and proof sketches; their proofs are not verified here. Vose's theorem is recorded from its zbMATH review and the restatements cited; the paper is not held. The status-defining proof is the release's Lean development, built and audited here as the claim page records; its prose proof is not independently reviewed, and no proof is compiled on this page. The Lean proof of the lower bound by Harmonic's Aristotle prover is not built here.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• doorn_2026_practical_numbers_egyptian_fractions
• doorn_2026_practical_numbers_egyptian_fractions / proposition_4_1
• doorn_2026_practical_numbers_egyptian_fractions / theorem_1_2
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• doorn_2025_smallest_denominator_not_contained_unit_fraction
• doorn_2025_smallest_denominator_not_contained_unit_fraction / section_3
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / theorem_1
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / theorem_2
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / theorem_4
• openai_2026_short_egyptian_fractions
• openai_2026_short_egyptian_fractions / corollary_1_3
• openai_2026_short_egyptian_fractions / theorem_1_1