Problem 306

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claims/: The 6 claim pages of Problem 306, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that

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

Formulation. The statement is the site's (page last edited 21 June 2026). The denominators are semiprimes with two distinct prime factors (the formal file writes $\omega(n_i)=2$ and $\Omega(n_i)=2$), and $b$ squarefree is necessary, since a square factor of $b$ cannot divide a common denominator that is a product of distinct primes. The case $b=1$ (every natural number) is a special case; the site's commentary reports the analogous statement for denominators with three distinct prime factors, which is a different denominator class. The 1980 monograph (printed p. 38) records that the authors had proved, unpublished, the three-prime statement for every $a/b$ with $b$ squarefree, and adds "Whether this can be done with two prime factors is not clear"; Butler, Erdős and Graham (2015, p. 2) say the three-prime rational statement was "attributed to two of the authors ... however a proof of this result was never published", prove the natural-number case, and "conjecture that a similar result holds for $\omega=2$".

Status. Open, in the site's label. The frontmatter standing is claimed with claim proved, derived from two pending full claims, each on its own page: Tang's Lean proof of June 2026, a Lean development asserting the whole statement with two Rosser--Schoenfeld bounds taken as axioms, followed by a manuscript published in October 2026, and Li's elementary proof of September 2026, a preprint with a Lean formalization and the one full proof claim on the site's tab. A third page, Li's earlier preprint of June 2026, is a partial claim covering the case $b=1$ and every $a/b$ above an explicit threshold. Three further partial pages record the representations of $1$ by two-prime denominators that settle the instance $a/b=1$, Barbeau's of 1977, Johnson's of 1978 and Watanabe's of 2020, each claimed. None of the six is refereed, independently reviewed or accepted by the site, which labels the problem OPEN (page last edited 21 June 2026, and the community database agreeing, as of 2026-10-07); this corpus has built none of the Lean developments. No refereed source proves the statement for any $a/b$ beyond finite examples for $a/b=1$; the refereed theorem on record is Butler, Erdős and Graham's Theorem 1, the natural-number case with three prime factors instead of two.

Source. erdosproblems.com/306, accessed 2026-09-18: the problem page (OPEN, with the site's note that the problem is not one a finite computation can settle; source key [ErGr80]; last edited 21 June 2026), its discussion thread (10 comments, 25 February 2026 to 21 June 2026) and its proof-claim tab, empty on that date and carrying one full claim with three comments on 2026-10-06 (the page itself unchanged). Cite as: T. F. Bloom, Erdős Problem #306, https://www.erdosproblems.com/306, accessed 2026-09-18.

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), p. 38 (the site gives no page). Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [BEG15] Butler, S., Erdős, P. and Graham, R., Egyptian fractions with each denominator having three distinct prime divisors. Integers 15 (2015), paper A51, 9 pp. (received 21 February 2015, accepted 17 November 2015). Theorem 1, p. 2; Theorem 3, p. 8. Library home: butler_2015_egyptian_fractions_each_denominator_having_three.
• [Ba77] Barbeau, E. J., Expressing one as a sum of distinct reciprocals: comments and a bibliography. Eureka (Ottawa) 3 (1977), no. 7 (August--September 1977), 178--181; the journal became Crux Mathematicorum in 1978, and the issue is in the Canadian Mathematical Society's archive, Crux_v3n07_Aug.pdf (accessed; the 101 denominators are printed on pp. 178--179). Claim page 1977_08_01_barbeau.
• [Jo78] Johnson, A. W., Jr., Letter to the editor. Crux Mathematicorum 4 (1978), no. 7 (August--September 1978), 190, in the same archive, Crux_v4n07_Aug.pdf; the 48 denominators are also reproduced in [BEG15], p. 2, and [Wa20], p. 2. Claim page 1978_08_01_johnson.
• [Wa20] Watanabe, T., New examples of the representation of 1 by the sum of reciprocals of semiprime numbers. arXiv:2009.03275v2 (9 September 2020; v1 1 September 2020), 11 pp.; preprint, no journal record found. Section 2, p. 2; Section 3.3, pp. 5--10; Section 4.1, p. 11. Library home: watanabe_2020_new_examples_representation_1_sum_reciprocals; claim page 2020_09_01_watanabe.
• [Li26] Li, S., Every natural number is a sum of distinct semiprime unit fractions. arXiv:2606.15159v2 (17 June 2026; v1 13 June 2026), 22 pp.; preprint. Theorem 1.2, p. 2; Theorem 7.1, p. 12; Theorem 7.4, p. 14; Theorem 8.1, p. 14; Corollary 8.6, p. 17; Section 9, p. 19; "Use of AI", p. 22. Claim page 2026_06_13_li.
• [Li26b] Li, S., Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306. arXiv:2609.32140v1 (26 September 2026), 13 pp., with a Lean 4 formalization as ancillary files and in the repository daizisheng/erdos-306; preprint, no journal record. Cited from its abstract and the repository's description; claim page 2026_09_26_li.
• [Ta26] Tang, Y., A machine-checked proof of Erdős Problem 306 in Lean 4 (v0.0.3). Zenodo, DOI 10.5281/zenodo.20767390 (published 19 June 2026; software), and the repository Yuren-Tang/erdos-306 (release v0.0.3 published 2026-06-19T21:34Z); with the manuscript Squarefree semiprime unit fractions: a characterization and a local limit theorem, frozen 11 August 2026 and published in the repository on 3 October 2026. Cited from its release metadata and the author's descriptions; this corpus has not built or read it. Claim page 2026_06_19_tang, which pins the release and the manuscript.

Formalization. Statement only. The file ErdosProblems/306.lean of formal-conjectures at the linked commit (main on 2026-09-18) declares erdos_306 : answer(sorry) ↔ ∀ (q : ℚ), 0 < q → Squarefree q.den → ∃ k : ℕ, ∃ (n : Fin (k + 1) → ℕ), n 0 = 1 ∧ StrictMono n ∧ (∀ i ∈ Finset.Icc 1 (Fin.last k), ω (n i) = 2 ∧ Ω (n i) = 2) ∧ q = ∑ i ∈ Finset.Icc 1 (Fin.last k), (1 : ℚ) / (n i) under category research open with proof sorry, and the variant erdos_306.variants.integer_three_primes (every positive integer with three distinct prime factors per denominator; category research solved, proof sorry, no formal_proof attribute). The community database records the problem as open (31 August 2025), the statement as formalized since 31 August 2025, and no formal proof or formal-proof URL. The database keeps that record and adds a formal-status note: a Lean formalization by Collin Yuanjie Ren, AI-assisted, of Watanabe's 47-term decomposition of $1$, the instance $a/b=1$ and not the general rational question, with a pinned link to its README; it is a formalization link on Watanabe's claim page. The Zenodo release [Ta26] describes its headline theorem as stated in the same form as the formal-conjectures proposition, and the repository of [Li26b] declares a theorem that its README describes as proving that proposition with the three standard axioms; both are the authors' own descriptions, recorded on the claim pages. This corpus has built and audited none of the three developments.

Current assessment

The question. The site's statement above; OPEN; last edited 21 June 2026. The commentary records that the three-prime analogue holds for $b=1$ by the theorem of Butler, Erdős and Graham [BEG15], a paper the site notes appeared nineteen years after Erdős's death and may be his last, and that several representations of $1$ with two-prime denominators are known, Barbeau's [Ba77] the first and Watanabe's 47-term examples [Wa20] the shortest, with the introduction of [Wa20] cited for the history. The thread (ten comments): 25 February 2026, a survey of examples for $a/b=1$ (Barbeau 1976/77 with 101 terms; Burshtein's constructions of 2006 and 2010, for example 52 terms with largest denominator 1963; OEIS A201650 for Johnson's 48-term example; Watanabe's 94 examples with 48 terms and 17 with 47), which the commenter credits to an AI model and which the site then used; 18 June 2026, the author of [Li26] announces the preprint, with a disclosure of human--AI collaboration and a proof summary, and the site's curator replies the same day that the preprint claims a partial result, many rationals including every integer, rather than a proof of the whole problem; two commenters report that the paper proves the $\omega=2$ case for all rationals above $1/5$ (more precisely above a threshold $T(r)$ depending on the largest prime factor $r$ of $b$, equal to $1/5$ for $r\ge59$) and the $\omega=3$ case for all rationals, one of them relaying an AI screening check of the informal proof (a chat transcript, which this page does not cite); 19--21 June 2026, the announcement of [Ta26], two replies and the author's follow-up of 20 June. The community database record (above) agrees with the label.

Origin. Printed p. 38 of the 1980 monograph: "The authors (unpublished) have proved that for any $\frac ab$ with $b$ squarefree there are infinitely many disjoint sets $S=\{s_1,\dots,s_r\}$ such that each $s_k$ is a product of three distinct prime factors and $\frac ab=\sum_{i=1}^r\frac1{s_k}$ [sic]. Whether this can be done with two prime factors is not clear." The same page records Barbeau's 101-term example of $1$ with two-prime denominators [Bar (77)] and Burshtein's earlier example with $x_i\nmid x_j$ [Burs (73)]. The site's question is the two-prime case of the monograph's remark.

What is proved (refereed sources). Theorem 1 of Butler, Erdős and Graham (p. 2): any natural number can be written as an Egyptian fraction where each denominator is the product of three distinct primes; Theorem 3 (p. 8): the same with three distinct odd primes. The proof (pp. 3--7) shows by induction that the subset sums of the products of $n$ of the first $n+3$ primes contain every integer between $1/6$ and $5/6$ of their total (Lemma 1), and divides by the product of the first $n+3$ primes. The paper states the two-prime statement as a conjecture (p. 2) and explains (p. 8) why its method does not reach small rationals, the ends of the interval being not understood. Examples for $a/b=1$ with two-prime denominators, each the instance $a/b=1$ of the question and each on its own partial claim page: Barbeau's 101 terms (1977; printed in [Ba77], pp. 178--179, with largest denominator $1838171$), Johnson's 48 terms (1978; the denominators $6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,62,65,69,77,82,85,86,87,91,93,95,115,119,123,133,155,187,203,209,215,221,247,265,287,299,319,323,391,689,731,901$ printed in [Jo78] and reproduced in [BEG15] p. 2 and [Wa20] p. 2), and Watanabe's seventeen examples with 47 terms (Section 2, p. 2; printed in Section 3.3, pp. 5--10), with no 46-term example when the prime factors are at most $101$ (Section 4.1, p. 11); the minimality of $47$ is the author's expectation, not a theorem. Exact rational arithmetic for this page confirms that Barbeau's 101 numbers, Johnson's 48 and the first of Watanabe's 47-term examples (largest prime factor $53$) are distinct products of two distinct primes whose reciprocals sum to exactly $1$; the other sixteen of Watanabe's examples are recorded as printed. An elementary connection (an author-recorded check): if finite sets of primes $P$ and $Q$ satisfy $(\sum_{p\in P}1/p)(\sum_{q\in Q}1/q)=1$, as Problem 307 asks, then $P\cap Q=\emptyset$ is forced and $1=\sum_{p\in P,q\in Q}1/(pq)$ is a representation of $1$ of this problem's form; no converse holds, and no such pair is known.

Claims (each on its own page; none accepted). The standing derives from these pages, which hold the details of each posting.

• Barbeau's 101-term representation of 1 (partial, claimed): [Ba77] prints 101 distinct products of two distinct primes whose reciprocals sum to $1$, the first such representation, found with a pocket calculator; it settles the instance $a/b=1$. The venue is a problem-solving journal and no refereeing of the article is evidenced, and the site's credit is commentary on a problem it labels OPEN, so the page lists no acceptance evidence.
• Johnson's 48-term representation of 1 (partial, claimed): [Jo78], a letter to the editor answering Barbeau's question for the fewest terms, prints the 48 denominators above; the same grounds leave it without acceptance evidence, although the example is reproduced in the refereed [BEG15].
• Watanabe's seventeen 47-term representations of 1 (partial, claimed): [Wa20] prints seventeen representations with 47 terms, the shortest known, and reports that none with 46 terms exists when the prime factors are at most $101$; preprint only. Collin Yuanjie Ren's AI-assisted Lean formalization of the first example, which the community database notes, is a formalization link on the page; this corpus has not built it.
• Li's semiprime representations of integers and large rationals (partial, claimed): [Li26] proves the case $b=1$ and every $a/b$ with $b$ squarefree and $a/b\ge\min\{B_{N_b}/6,1/5\}$, where $B_N=\sum_{i<j\le N}1/(p_ip_j)$ and $N_b$ is the index of the largest prime factor of $b$, by adapting the induction of [BEG15]; its Section 9 reduces the rationals below the threshold to a conjecture on the gap-free floor of the semiprime subset-sum set. The preprint also proves the rational statement for denominators with exactly $k$ distinct prime factors for every $k\ge3$, a different denominator class. Provenance declared by the source: a human--AI collaboration with a Lean 4 development in the arXiv source, which this corpus has not built.
• Tang's Lean proof of the full statement (full, claimed): [Ta26], a Lean 4 development whose record describes its headline theorem as the formal-conjectures proposition, complete apart from two Rosser--Schoenfeld (1962) bounds stated as axioms, announced in the discussion on 19 June 2026; the author's manuscript of August 2026 was published in October 2026 with notice that the proof has since been simplified. This corpus has not built, read or audited it.
• Li's elementary proof of the full statement (full, claimed): [Li26b] asserts the whole statement by counting, through a finite Fourier sum, the subgraphs of one complete bipartite graph of primes whose reciprocal sum is congruent to $a/b$ modulo $1$, with only Chebyshev-type bounds on primes as input; it credits [Ta26] with the first proof and reuses its framework. The repository's Lean development reports the three standard axioms for a theorem proving the formal-conjectures proposition, which this corpus has not built. This is the one full proof claim on the site's tab.
• Discussion, 25 February 2026: the compilation of examples credited to an AI model was folded into the site's commentary (Barbeau, Watanabe). It claims no result and has no page. Burshtein's constructions (2006, 2010) and OEIS A201650 are recorded only from that comment, with no source of theirs cited on this page, so they have no page.

Search scope. The status rests on these routes; none found a refereed proof or an accepted resolution.

• The site: problem page, discussion thread, empty proof-claim tab; the community database record; formal-conjectures 306.lean at the pinned commit.
• The primary sources read as stated: [ErGr80] (p. 38), [BEG15] (pp. 2 and 8, with pp. 3--7 for structure), [Wa20] (pp. 2 and 11, and pp. 3--10), [Li26] (as listed above).
• Publication records: arXiv listings of 2009.03275 (v2, no journal reference) and 2606.15159 (v1, v2, no journal reference); Crossref bibliographic queries for [Wa20] and [Li26] (no records); Semantic Scholar for [Li26] (record present, zero citations).
• arXiv API metadata search for semiprime or two-distinct-prime denominators with unit or Egyptian fractions (two records: [Li26] and arXiv:2602.02012 on denominators $p^aq^b$); the Zenodo API for record 20767390; the GitHub API for Yuren-Tang/erdos-306 (repository, releases, head commit).
• Later accesses: the site's proof-claims tab and its comments (2026-10-06); the arXiv listing of 2609.32140 (v1 only), the Zenodo record and the release and commit records of both repositories, the Society's archive copies of Eureka 3, no. 7 and Crux Mathematicorum 4, no. 7, the arXiv record of 2009.03275, the community database (with its formal-status note) and Ren's README at the pinned commit (2026-10-07).

Not searched: MathSciNet, zbMATH, Google Scholar full text, X. Not examined: Burshtein's papers, Guy's problem book (D11), the Lean sources of [Li26], [Ta26] and Ren.

Remaining gaps. (1) Six claims, four partial (the three representations of $1$ and Li's June preprint) and two asserting the whole statement, with no refereed publication, independent review or site acceptance, and three Lean developments, none built in this corpus. (2) The three-prime rational statement the monograph calls proved (unpublished) has no refereed proof; [Li26] claims one. (3) The representations of $1$ by Barbeau, Johnson and Watanabe are checked by exact arithmetic on this page, an author-recorded check; Burshtein's constructions are known only from the thread. (4) Nothing refereed addresses the two-prime statement beyond finite examples for $a/b=1$. There is no status-defining proof to compile.

Proof coverage. Nothing here is accepted; the claimed standing rests on the two pending full claims. Butler, Erdős and Graham's Theorem 1 is recorded at statement level with its proof read for structure; the three representations of $1$ are finite identities checked by exact arithmetic; the three 2026 claims are recorded at the level of their statements, abstracts, declarations and release metadata. No proof has been rewritten or independently reviewed.

Proof claims on the site. The site's proof-claims tab carries one full proof claim, submitted 2026-09-29 by Shisheng Li, who names GPT-5.6-sol, GPT-6-astra and Claude Opus 5.5 as the systems used, with [Li26b] as the write-up and its repository as the formalization; its summary asserts the whole statement and sketches the construction recorded on its claim page. Its three comments (29 September to 3 October 2026) are an exchange with Tang, who welcomes the proof, reports simplifying his own and publishes his August manuscript (recorded on his claim page); none is a review. The site labels the problem OPEN (page last edited 21 June 2026, as of 2026-10-07).

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.

• erdos_1980_old_new_problems_results_combinatorial_number_theory
• butler_2015_egyptian_fractions_each_denominator_having_three
• butler_2015_egyptian_fractions_each_denominator_having_three / theorem_1
• watanabe_2020_new_examples_representation_1_sum_reciprocals
• watanabe_2020_new_examples_representation_1_sum_reciprocals / section_2