Problem 293

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claims/: The 3 claim pages of Problem 293, one per claimant's result; the problem's standing derives from them.

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Statement. Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to

$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$

with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$.

Statement (corrected). Let $k\geq 1$ and let $v(k)$ be the minimal integer $>1$ which does not appear as some $n_i$ in a solution to

$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$

with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$.

Notes. The site's $v(k)$ is degenerate for every $k\ge2$. The integer $1$ occurs as a denominator only in the one-term solution $1=1/1$, since a solution with $k\ge2$ terms that contains $1/1$ sums to more than $1$; so for every $k\ge2$ the least positive integer in no solution is $1$, and over all integers there is no least one. Either way there is no growth to estimate, at every $k\ge2$ and not only at boundary values. The change inserts "$>1$" after "minimal integer"; nothing else changes. The evidence is the poser's own statement of the question: the 1980 monograph [ErGr80], printed p. 35, asks for "the least integer $v(n)>1$ which does not occur as an $x_k$". The site's own commentary reads the same quantity, since its $v(k)\gg k!$ and van Doorn and Tang's $v(k)\ge e^{ck^2}$ are false of the site's wording; van Doorn and Tang define $v(k)$ as the least integer above $1$ missing from every $k$-term solution, a comment of 8 December 2025 in the site's discussion clarifies the definition the same way, and the community database marks the statement "ambiguous statement". The defect is the site's: the monograph prints the condition. No result concerns the site's wording beyond the observation above, which is this corpus's own and counts for nothing. The standing judges the corrected Statement.

Formulation. Writing $D_k$ for the set of denominators that occur in some $k$-term representation of $1$ by distinct unit fractions, $v(k)=\min\{m>1:m\notin D_k\}$. Van Doorn and Tang's Lemma 2.1 records $D_2=\emptyset$ and $D_3=\{2,3,6\}$, and $D_1=\{1\}$, so $v(1)=v(2)=2$ and $v(3)=4$.

Status. Open on the site: the label is OPEN, which the commentary attaches to the corrected Statement (page last edited 29 December 2025; no proof claim on its tab; read 2026-10-07), and the standing derived from the claim pages is open, since every claim is partial. The growth of $v(k)$ is fixed at the double-logarithmic scale and not beyond: for all large $k$, $e^{e^{k/600}}\le v(k)$ and $\log\log v(k)=\Theta(k)$ with $\liminf\log\log v(k)/k\ge\log2/257$ and $\limsup\le\log2$, by Corollary 1.3 of the OpenAI mathematics release's manuscript of 25 September 2026, accepted as a partial claim on formalized evidence (its claim page). The best upper bound remains $v(k)\le c_0^{(2/5+o(1))2^k}$ with $c_0=1.26408\ldots$ the Vardi constant, from van Doorn and Tang's inequality (1.2) and the Elsholtz–Planitzer count (the paper prints $(1/5+o(1))2^k$; see the Upper bound below); whether the slope tends to $\log2$ and any asymptotic for $v(k)$ are open. Before the release the best lower bound was $v(k)\ge e^{ck^2}$ (van Doorn and Tang, Theorem 1.1, for every $k\ge1$), accepted as a refereed partial claim (its claim page); a claimed intermediate bound, doubly exponential in $\sqrt k$, has its own page (van Doorn and GPT-6 Astra Pro, 16 September 2026).

Source. erdosproblems.com/293, read 2026-09-17: the problem page (OPEN; last edited 29 December 2025), its four-comment discussion thread and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #293, https://www.erdosproblems.com/293, 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. 35.
• [BlEr75] Bleicher, M. N. and Erdős, P., The number of distinct subsums of $\sum_{i=1}^N 1/i$. Math. Comp. 29 (1975), 29--42.
• [BlEr76] Bleicher, M. N. and Erdős, P., Denominators of Egyptian fractions. J. Number Theory 8 (1976), 157--168; and Denominators of Egyptian fractions II. Illinois J. Math. 20 (1976), 598--613. Cited with [BlEr75] by the monograph for the $k!$ claim; part II has its own card, Denominators of Egyptian fractions II.
• [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 (v1 26 December 2025; v2 24 May 2026); Math. Proc. Cambridge Philos. Soc., published online 8 July 2026, doi:10.1017/S0305004126102102.
• [Er50c] Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210; Theorem 3, p. 197.
• [ElPl21] Elsholtz, C. and Planitzer, S., Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. 53 (2021), 695--709; Corollary 3.
• [Vo85] Vose, M. D., Egyptian fractions. Bull. London Math. Soc. 17 (1985), 21--24. The input of the lower bound, used as restated in [vDTa25b] (Lemma 2.2).

Formalization. No formal-conjectures statement: no file ErdosProblems/293.lean existed on 2026-09-17, and the pull request of 4 October 2026 that would add one formalizes the setting only and says it is not a solution. The community database (fetched and 2026-10-06) records the problem as unformalized with no formal-proof URL, and the site's status label carries no Lean suffix. The release's Lean proof of the partial result, Corollary 1.3, built and audited by the corpus's verification, is recorded on its claim page; the author-side Lean file of the van Doorn note is a formalization link on that note's claim page and is not built.

Current assessment

The question. On 2026-09-17 the site states the problem as above, shows OPEN, cites [ErGr80, p. 35], attributes $v(k)\gg k!$ to [BlEr75], records the elementary bound $v(k)\le kc_0^{2^k}$ with $c_0=1.26408\ldots$, and records van Doorn and Tang's $v(k)\ge e^{ck^2}$ together with their connection to Problem 304. The proof-claim tab is empty; the discussion has four comments, described below. The monograph's p. 35 (Erdős–Graham 1980) asks for "the least integer $v(n)>1$ which does not occur as an $x_k$", says "It is easy to see that $v(n)>cn!$ using results of Bleicher and Erdős" with all three papers cited, and adds that $v(n)$ "may" grow "more like $2^{2^{\sqrt n}}$ or $2^{2^{n(1-\varepsilon)}}$".

Claims. Three claim pages, all partial, so the derived standing is open. The first, van Doorn and Tang's Theorem 1.1 (26 December 2025), status accepted on refereed evidence, claim proved: $v(k)\ge e^{ck^2}$ for every $k\ge1$, with the upper bound $v(k)\le c_0^{(2/5+o(1))2^k}$ from their inequality (1.2) and the Elsholtz–Planitzer count. The second, the van Doorn note's Theorem 1.3 (16 September 2026), status claimed, claim proved: $v(k)\ge\exp\exp\sqrt{k/(2c_0)}$ with $c_0=14/\log2$ for all large $k$, described below. The third, the OpenAI release's Corollary 1.3 (25 September 2026), status accepted, scope partial, claim proved: $e^{e^{k/600}}\le v(k)\le 1+k^{2^{k-1}}$ for all large $k$ and $\log2/257\le\liminf\log\log v(k)/k\le \limsup\log\log v(k)/k\le\log2$. The page states what is and is not covered, 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. Its lower bound replaces the published $e^{ck^2}$ for large $k$, implies the claimed doubly exponential bound of the van Doorn note and the superseded $\exp(k^A)$ deduction below, and rules out the monograph's $2^{2^{\sqrt n}}$ guess; its upper bound is weaker than the recorded one.

Bounds. The known bounds are two-sided: $e^{e^{k/600}}\le v(k)\le c_0^{(2/5+o(1))2^k}$ for all large $k$, with $c_0=1.26408\ldots$ the Vardi constant, so $\log\log v(k)=\Theta(k)$ with a gap of a constant factor in the slope, between $\log2/257$ and $\log2$.

A claimed weaker bound, not adopted. A mostly AI-generated note, van Doorn 2026 (by van Doorn and GPT-6 Astra Pro, with ChatGPT and Aristotle named for the argument and its Lean file; claim page van Doorn, 16 September 2026) claims, as its Theorem 1.3, that every integer between $2$ and $\exp\exp\sqrt{k/(2c_0)}$, $c_0=14/\log 2$, occurs as a denominator in some $k$-term decomposition of $1$ for large $k$; the note is unrefereed, its author-side Lean formalization has not been built by the corpus, and the claim is not adopted into the bounds above; the release's accepted bound exceeds it for large $k$ and so implies it.

Site and database state (read 2026-10-05 and 2026-10-06). The site shows OPEN (last edited 29 December 2025), five comments and no proof claim on this problem; the van Doorn claim above sits on the proof-claims tab of Problem 18 (submitted 2026-09-16, no comments, not accepted, not on arXiv); the community database lists the problem open and unformalized; formal-conjectures has no file (its open pull request 6842 of 4 October 2026 formalizes the setting only and says it is not a solution); and there is no conjectures.io or Palomar entry. New finite data (a lead, unverified): the comment of 23 September 2026 (the user JudeWallis) gives $v(9)=13{,}856{,}993$, with certificates for every $2\le m\le13{,}856{,}992$ and an exhaustive search ruling out $13{,}856{,}993$ (repository EconLearn/erdos293-check, AI-assisted); OEIS A400352, approved about 1 October 2026, lists $v(3),\dots,v(9)=4,11,17,103,733,27539,13856993$ (community database issue 403 and pull request 452). This supersedes the bound $v(9)>2{,}108{,}538$ below and does not bear on the growth question.

• Lower bound before the release, and the best bound that holds for every $k\ge1$ (the release's accepted bound has an unspecified threshold). van Doorn–Tang, Theorem 1.1 (claim page van Doorn and Tang, 26 December 2025): there is an absolute $c>0$ with $v(k)\ge e^{ck^2}$ for all $k\ge1$. The paper is refereed (Mathematical Proceedings of the Cambridge Philosophical Society, online 8 July 2026, per its Crossref record); the statements are those of arXiv v2, and the published text has not been compared with it. The proof rests on the nesting $D_k\subseteq D_{k+1}$ (Lemma 2.1) and on Vose's theorem that every $a/b\in(0,1)$ is a sum of at most $C\sqrt{\log b}$ distinct unit fractions with denominators of a special form (Lemma 2.2, restated from [Vo85]). The authors write that no lower bound existed in the literature before theirs and that extracting the monograph's $k!$ from the Bleicher–Erdős papers "does not seem straightforward" to them.
• The $k!$ attribution. The 1975 paper cited by the site proves lower bounds for the number $S(N)$ of distinct subsums of $\sum_{i\le N}1/i$ and contains no statement about the denominators of $k$-term representations of $1$ (see its card). The two 1976 papers concern the largest denominator $D(N)$ and, in part II, $S(N)$ again (part I card, part II card). The $k!$ line is therefore the monograph's claim, not a theorem stated in the sources it cites; no reconstruction of it is compiled, and Theorem 1.1 supersedes it.
• Upper bound. Inequality (1.2): $v(k)\le|D_k|+2\le kF(k)+2$, where $F(k)$ counts the $k$-term representations (Problem 148), and the Elsholtz–Planitzer bound on $F(k)$ gives $v(k)\le c_0^{(2/5+o(1))2^k}$: Elsholtz and Planitzer's Corollary 3(2) (arXiv v1, p. 5) bounds the count by $c^{(2/5+\varepsilon)2^{k-1}}$ with $c=1.5979\ldots=c_0^2$. The paper, like the site's commentary for Problem 148, prints $c_0^{(1/5+o(1))2^k}$, which pairs that exponent with the Vardi constant and states a bound the cited corollary does not give. The site's cruder $v(k)\le kc_0^{2^k}$ and the discussion's $v(k)\le s_k$ for Sylvester's sequence $s_1=2$, $s_{j+1}=s_j(s_j-1)+1$ both come from the fact that every denominator of a $k$-term representation of $1$ is at most $s_k-1$, which is Erdős's Theorem 3 of 1950 (the extremal solution is $2,3,7,\ldots,s_{k-1},s_k-1$).

Connection to Problem 304. Section 3 of the paper: if $b<v(k)$ then $b\in D_k$, and removing $1/b$ from a $k$-term representation of $1$ gives $N(b-1,b)\le k-1$, so lower bounds for $v$ give upper bounds for $N(b-1,b)$; conversely the authors write that if the conjecture $N(b)\ll\log\log b$ of Problem 304 holds, "it seems likely" that their method gives $v(k)\ge e^{e^{ck}}$. The first direction is a two-line argument; the second is a stated expectation that the release's Corollary 1.3 realizes by its own route (the claim page above). Neither changes the status. Erdős's own proof of the lower bound $N(b-1,b)>\log\log b-1$ (Theorem 2 of 1950) already runs through the same link.

Finite values (unreviewed leads). No refereed source tabulates $v(k)$. Two AI-assisted computations are recorded here as leads with their own disclosures, not as results; the site warns that comments are not verified, and nothing below is verified in this corpus.

• Site discussion, comment of 14 September 2026 by the user islomifaridun: $v(3),\ldots,v(8)$ as $4,11,17,103,733,27539$ and $v(9)>2108538$, with programs and certificates in a GitLab repository (gitlab.com/faridunislom/erdos293, created 11 September 2026; its landing page was reachable on 2026-09-17 and its contents were not examined). The comment discloses AI assistance and allows that $v(6)$ and $v(7)$ were perhaps already known.
• A public AI-assisted report dated 26 July 2026 (erdosproblemaday.com/report/293) labels itself partial and computational-only, gives $v(1),\ldots,v(7)=2,2,4,11,17,103,733$ and $2307\le v(8)\le27539$, and says the asymptotic question remains open. The two sources agree where they overlap.

Search scope. The status rests on the following routes; none found a proof, disproof, preprint or claim of the growth rate.

• The site: problem page, discussion (four comments: the AI-assisted computation of 14 September 2026 described above, the comment of 29 December 2025 by the paper's second author announcing the result, his comment of 8 December 2025 deriving $v(k)\le s_k$ from the Sylvester sequence, and the comment of 8 December 2025 that states inequality (1.2) and clarifies the definition; the site text reflects the last three) and the empty proof-claim tab; the community database record (open, unformalized); the formal-conjectures directory (no file on that date).
• arXiv: the abstract page of 2512.22083 (v1, v2 and the acceptance comment) and API metadata searches for "unit fraction" AND "smallest denominator" (no record), "unit fractions" AND denominators AND distinct (eleven records, none on $v(k)$), Bleicher AND Egyptian (one record, on $S(N)$) and a sweep of 2025–2026 abstracts mentioning "Egyptian fractions" or "unit fractions" (29 records, none on $v(k)$). Titles and abstracts only.
• Crossref: the journal record of [vDTa25b]. Semantic Scholar: the citation list of arXiv:2512.22083 (empty on the search date). zbMATH Open: au:Bleicher & au:Erdős & ti:Denominators (the two 1976 papers).
• The primary sources: [BlEr75], part I of [BlEr76] and pp. 598–600 and 602–603 of part II, [Er50c], [vDTa25b] (v2) and [ErGr80] p. 35.

Not searched: MathSciNet, Google Scholar, X. [Vo85] was not consulted; its theorem is used as restated in [vDTa25b] (Lemma 2.2).

Proof coverage. Theorem 1.1 is paged as statement, locator and sketch (claims checked); neither its proof nor Vose's theorem is reconstructed or independently reviewed in this corpus. The release's partial claim rests on its Lean development, built and audited by the corpus's verification as its page records, with its prose proof read for structure only on the source card. The van Doorn note's claim is unverified. The status is a two-sided estimate, not a resolution, so there is no status-defining proof to compile.

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_3
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• bleicher_1975_number_distinct_subsums_sum_n_1
• bleicher_1976_denominators_egyptian_fractions_ii
• doorn_2025_smallest_denominator_not_contained_unit_fraction
• doorn_2025_smallest_denominator_not_contained_unit_fraction / inequality_1_2
• doorn_2025_smallest_denominator_not_contained_unit_fraction / section_3
• doorn_2025_smallest_denominator_not_contained_unit_fraction / theorem_1_1
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / theorem_4
• openai_2026_short_egyptian_fractions
• openai_2026_short_egyptian_fractions / corollary_1_2
• openai_2026_short_egyptian_fractions / corollary_1_3
• openai_2026_short_egyptian_fractions / theorem_1_1