Problem 305

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claims/: The 4 claim pages of Problem 305, one per claimant's result; the problem's standing derives from them.

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Statement. For integers $1\leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with

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

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

$$D(b) \ll b(\log b)^{1+o(1)}?$$

Formulation. $D(a,b)$ depends only on the rational $a/b$. Bleicher and Erdős write $D(N)=\max_{0<a<N}D(a,N)$, and the question is their Conjecture 3 of 1976: for every $\varepsilon>0$ there is $K(\varepsilon)$ with $D(N)\le K(\varepsilon)N(\ln N)^{1+\varepsilon}$; the site's $D(b)\ll b(\log b)^{1+o(1)}$ says the same.

Status. Proved, in the site's label, PROVED (page last edited 18 November 2025). The answer is yes. Yokota (1988), in the paper the site cites as the solution, proved $D(N)/N\le(\log N)^{1+\delta(N)}$ with $\delta(N)\to0$ (the zbMATH review, Zbl 0652.10015), which Liu and Sawhney restate as $D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2$; Liu and Sawhney (2024; published 2026) improved this to $D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$. Either bound implies $b(\log b)^{1+o(1)}$. Yokota's paper is refereed but paywalled, so its statement is recorded from the zbMATH review. Liu and Sawhney's Theorem 1.5 is refereed (International Mathematics Research Notices, published online 14 January 2026); the library records it from arXiv v1 as a statement with a proof sketch. Two accepted full claims carry the standing: Yokota's 1988 theorem, the site's citation, and Liu and Sawhney's Theorem 1.5. Two accepted partial claims record Bleicher and Erdős's bounds: their J. Number Theory paper ($D(P)\ge P\lceil\log_2P\rceil$ for primes and $D(N)\le KN(\ln N)^3$) and their Illinois paper ($D(N)\le\lambda^3(N)N(\ln N)^2$ and a sharper prime lower bound). The lower bound $D(p)\gg p\log p$ for primes shows that the exponent $1$ of $\log b$ cannot be lowered. The site's discussion and proof-claim pages carry no further claim.

Source. erdosproblems.com/305: the problem page (PROVED, with the site's note that it is solved in the affirmative; last edited 18 November 2025), its empty discussion thread and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #305, https://www.erdosproblems.com/305, 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), p. 38.
• [BlEr76] Bleicher, M. N. and Erdős, P., Denominators of Egyptian fractions. J. Number Theory 8 (1976), 157--168 (the site's reference reads "Denominators of unit fractions"); Theorem 1, p. 158; Theorem 2, p. 162; Conjecture 3, p. 167.
• [BlEr76b] Bleicher, M. N. and Erdős, P., Denominators of Egyptian fractions II. Illinois J. Math. 20 (1976), 598--613; Theorem 1, p. 602.
• [Yo88] Yokota, H., On a problem of Bleicher and Erdös. J. Number Theory 30 (1988), no. 2, 198--207, doi:10.1016/0022-314X(88)90017-0. Paywalled; zbMATH review Zbl 0652.10015.
• [Yo86], [Yo88b] Yokota, H., On a conjecture of M. N. Bleicher and P. Erdős. J. Number Theory 24 (1986), 89--94; Denominators of Egyptian fractions. J. Number Theory 28 (1988), 258--271. The two earlier papers of the series, as cited by Liu and Sawhney; paywalled, with no open copy found (arXiv, zbMATH, publisher). They are historical context, not the status source.
• [LiSa24] Liu, Y. P. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113v1 (10 April 2024); Int. Math. Res. Not. IMRN 2026, no. 2, rnaf382, published online 14 January 2026, doi:10.1093/imrn/rnaf382. Theorem 1.5, p. 3 of v1.

Formalization. A statement file and an external Lean proof, neither built in this corpus. The statement file was added on 20 September 2026: at the linked commit, FormalConjectures/ErdosProblems/305.lean states erdos_305 (the answer is yes: there are $C>0$ and $\delta(b)\to0$ with $D(b)\le Cb(\log b)^{1+\delta(b)}$ for all large $b$) by sorry with a formal_proof annotation pointing at src/latest/ErdosProblems/Erdos305.lean in Boris Alexeev's lean-proofs collection at the linked commit of 15 September 2026, and the Bleicher–Erdős bound, the prime lower bound, Yokota's bound and Liu and Sawhney's bound as further sorry variants. As of 2026-10-07 the site's page shows the statement as formalized and links the statement file, and the community database records the problem as formalized since 20 September 2026; the site's status label carries no Lean suffix. The top file of the Alexeev development declares itself a Lean formalization of the affirmative resolution of Problem 305 with informal authors Bleicher, Erdős, Yokota, Liu and Sawhney and formal authors Codex, GPT-5.6 Sol (OpenAI Codex), citing [Yo88] and [LiSa24] among its primary references; its theorem erdos_305 proves the $b(\log b)^{1+o(1)}$ statement with constant $8$, not either paper's iterated-logarithm bound, and is linked as a formalization from both claim pages. This corpus has built neither file, so no formalized evidence is listed.

Current assessment

The question. The site states the problem as above, shows PROVED, cites [ErGr80, p. 38], and says: Bleicher and Erdős [BlEr76] showed $D(b)\ll b(\log b)^2$ and $D(p)\gg p\log p$ for primes; it credits the solution to Yokota [Yo88], with the bound $D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2$, and records Liu and Sawhney's [LiSa24] sharper bound $D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$. The monograph's p. 38 (Erdős–Graham 1980) states the same bounds, citing its "[Bl-Er (76) a]", which its bibliography (p. 108) identifies as the J. Number Theory paper, and the conjecture $D(b)\le c(\varepsilon)b(\log b)^{1+\varepsilon}$ for every $\varepsilon>0$.

What the Bleicher–Erdős papers prove. Part I: Theorem 1 (p. 158), $D(P)\ge P\lceil\log_2P\rceil$ for every prime $P$, the site's $D(p)\gg p\log p$; Theorem 2 (p. 162), $D(N)\le KN(\ln N)^3$ for all $N\ge2$; Conjecture 3 (p. 167), the question itself. Part II, Theorem 1 (p. 602): $D(N)\le\lambda^3(N)N(\ln N)^2$ for every $N$, with $2/\log2\ge\lambda(N)\ge1$ and $\lambda(N)\to1$; part II also sharpens the prime lower bound (its Theorem 4, p. 612). So the exponent-2 bound that the monograph, the site and Liu–Sawhney attribute to the J. Number Theory paper is the Illinois paper's theorem, while the J. Number Theory paper prints exponent 3 and part II's introduction recalls it with exponent 4. The site's commentary is not itself a source for any of these bounds; the theorem pages are. Part II has its own card, Denominators of Egyptian fractions II (Theorem 1, p. 602; Theorem 4, p. 612).

The solution. Liu and Sawhney (v1, p. 3) recount Yokota's series: a reduction to prime denominators [Yo86], the bound $D(a,b)\le b(\log b)^{3/2}$ [Yo88b], and finally [Yo88] with the iterated-logarithm bound quoted by the site; the site's solving citation is [Yo88]. Its Crossref record identifies the paper as J. Number Theory 30, no. 2 (October 1988), 198–207, and Semantic Scholar lists five citing works, among them a 1991 J. Number Theory paper on a problem of Erdős and Graham and the 2025 Bettin–Grenié–Molteni–Sanna paper on counting Egyptian fractions; none of the five improves the bound for $D(b)$. The paper is paywalled, and no open copy was found (arXiv search Yokota AND Egyptian: no record; zbMATH lists the 1986 and 1988 papers without links to copies; the publisher offers none). The statement is the one the zbMATH review (Zbl 0652.10015, by Ke Zhao) gives: $D(N)/N\le(\log N)^{1+\delta(N)}$ with $\delta(N)\to0$, establishing the Bleicher–Erdős conjecture; the explicit iterated-logarithm form above is Liu and Sawhney's restatement (arXiv:2404.07113v1, p. 3).

Liu–Sawhney, Theorem 1.5: for integers $1\le a<b$ there are integers $1<n_1<\cdots<n_k\le b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$ with $a/b=\sum1/n_j$. The paper is refereed (IMRN 2026, by its Crossref record); the library records the statement of arXiv v1 with a proof sketch through the paper's Lemma 4.1 and Proposition 3.2, and has not compared the published version. On the same page the authors attribute $D(a,b)\gg b(\log b)(\log\log b)^{1-o(1)}$ to part II, which is consistent with the exponent $1$ being sharp.

Search scope. Routes; none found a retraction, a dispute, or a later improvement of the bound.

• The site's three pages (no comments, no claims); the community database (proved, unformalized); formal-conjectures (no file).
• arXiv: the abstract page of 2404.07113 (v1 only, 22 pages); API metadata searches for "Egyptian fraction" AND "largest denominator" (two records, Martin's two 1998 preprints on dense and denser Egyptian fractions), Yokota AND Egyptian (none), Bleicher AND Egyptian (one, on counting subsums) and a sweep of 2025–2026 abstracts mentioning "Egyptian fractions" or "unit fractions" (29 records, none on $D(b)$).
• Crossref: the records of [Yo88] and [LiSa24]. Semantic Scholar: the five works citing [Yo88]. zbMATH Open: au:Yokota & ti:Bleicher (the 1986 and 1988 papers) and ti:"Egyptian fractions" & any:denominator & py:2020-2026 (one record, on ternary fractions with prime denominator).
• The primary sources: part I (all twelve pages); part II (pp. 598–600, 602–603); [LiSa24] v1 p. 3; [ErGr80] pp. 38 and 108.
• Also read: the formal-conjectures statement file and the Alexeev Lean file at the commits the Formalization links pin, the community database record (formalized since 20 September 2026) and the site's formalization panel.

Not searched: MathSciNet, Google Scholar, X. Not consulted: the texts of [Yo86], [Yo88b] and [Yo88] (paywalled; [Yo88] through its zbMATH review), and the published version of [LiSa24].

Proof coverage. Theorems 1 and 2 and Conjecture 3 of part I are paged (claims checked; no proof checked), and the two Bleicher–Erdős papers have accepted partial claim pages. Liu and Sawhney's Theorem 1.5 is a statement-and-sketch page; its full proof at the theorem's parameters is not compiled, and Yokota's proof is not compiled. The status rests on the refereed acceptance of the two solving papers, not on local proof coverage, and no independent review of either proof is recorded in this corpus.

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
• bleicher_1976_denominators_egyptian_fractions
• bleicher_1976_denominators_egyptian_fractions / conjecture_3
• bleicher_1976_denominators_egyptian_fractions / theorem_1
• bleicher_1976_denominators_egyptian_fractions / theorem_2
• bleicher_1976_denominators_egyptian_fractions_ii
• liu_2024_further_questions_regarding_unit_fractions
• liu_2024_further_questions_regarding_unit_fractions / theorem_1_5