Problem 292

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claims/: The 1 claim page of Problem 292, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A$ be the set of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$. Explore $A$. In particular, does $A$ have density $1$?

Formulation. The site's wording on 2026-09-17 (page last edited 20 December 2025). $A$ is the set of integers that occur as the largest denominator of a representation of $1$ by distinct unit fractions; $1\in A$ by the one-term representation, and for $n\ge2$ a representation with largest denominator $n$ has all denominators at least $2$. It is OEIS A092671 ($1,6,12,15,18,20,24,28,30,33,\ldots$). In Martin's notation, $A\setminus\{1\}$ is the complement of $\mathcal L_1(1)$, the set of integers greater than $1$ that cannot be the largest denominator. "Density $1$" is asymptotic density; the site's commentary states the finer order of the complement $B=\mathbb N\setminus A$.

Status. Proved, in the site's label, and the answer is yes: Martin's Theorem 4 (Acta Arith. 95 (2000), no. 3, 231--260; refereed) shows that $\mathcal L_1(r)$ has zero density for every positive rational $r$, with counting function of exact order $x\log\log x/\log x$; at $r=1$ this is $|B\cap[1,x]|\asymp x\log\log x/\log x$, so $A$ has density $1$. This is the accepted claim Martin 2000, refereed and credited by the site's curator.

Source. erdosproblems.com/292, accessed 2026-09-17: the problem page (PROVED, the site stating that the answer is yes; source key [ErGr80, p. 35]; last edited 20 December 2025), its empty discussion thread and its empty proof-claim tab. The site cites [Ma00] in its commentary and thanks Zach Hunter, Wouter van Doorn and Desmond Weisenberg. Cite as: T. F. Bloom, Erdős Problem #292, https://www.erdosproblems.com/292, accessed 2026-09-17.

References.

• [Ma00] Martin, Greg, Denser Egyptian fractions. Acta Arith. 95 (2000), no. 3, 231--260, DOI 10.4064/aa-95-3-231-260; arXiv:math/9811112v1 (18 November 1998, the only arXiv version, 26 pages; no file is held). Theorems 3 and 4, p. 3 of the preprint; proofs in Sections 6 and 7. Library home: martin_2000_denser_egyptian_fractions.
• [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. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [OEIS] Alekseyev, M., Sequence A092671, The On-Line Encyclopedia of Integer Sequences (2004; entry last modified 5 November 2025, server time): the elements of $A$ to $10000$ (b-file by J. E. Schoenfield), the observations that no prime power lies in $A$ and that multiples of elements greater than $1$ lie in $A$, and a conjectured characterization verified to $2\cdot10^5$; accessed.

Formalization. Statement in ErdosProblems/292.lean of formal-conjectures, added on 22 September 2026: it declares erdos_292 : answer(True) ↔ A.HasDensity 1 under category research solved with a sorry body and a formal_proof attribute pointing to the file src/latest/ErdosProblems/Erdos292.lean of the collection plby/lean-proofs, which names Martin as its informal author and Codex and GPT-5.6 Sol as its formal authors. The community database lists the problem as formalized as of its last update, of 22 September 2026. The external file is a formalization link on the claim page; nothing was built or audited by this corpus, so no formalized evidence is listed.

Current assessment

The question (site formulation of 2026-09-17). The statement above; status PROVED, last edited 20 December 2025; source key [ErGr80, p. 35]. The site's commentary records three facts about $A$: Straus's observation that the product of two elements of $A$ lies in $A$; the elementary exclusion of every prime power from $A$; and Martin's theorem [Ma00], which the site states as the affirmative answer together with the order $\log\log x/\log x$ for the relative size of $B=\mathbb N\setminus A$ up to $x$ and a description of $B$ as the small multiples of prime powers. It adds van Doorn's remark that $2n\in A$ whenever $n\in A$ and $n>1$, since halving a representation and adding the term $\frac12$ gives another. The thread and the proof-claim tab are empty. The community database, recorded proved, not formalized, OEIS A092671; as of its last update, of 22 September 2026, it lists the statement as formalized (see Formalization).

Origin. Printed p. 35 of the 1980 monograph: "What are the possible values of $x_n$ as $\{x_1,\ldots,x_n\}$ ranges over $\mathscr X$? As noted by Straus, the set of $x_n$ is closed under multiplication. Is it true that $x_n$ assumes almost all integer values? Note that $x_n$ is never a prime power, in fact $x_n\ne ap^k$ if $p$ is a prime exceeding $a!\log a$." The site's "Explore $A$" gathers the page's further questions (the least integer $v(n)$ that never occurs as an $x_k$, and the least integer $k_r(n)$ that never occurs as the $r$th-smallest denominator $x_r$ when all denominators are at most $n$). The analogue for the second-largest and later denominators, which Martin's Theorem 3 treats, is not posed on this page; Martin (p. 3) says it is mentioned in Guy's Unsolved problems in number theory.

Status support. The status-defining source is Martin's Theorem 4 (arXiv:math/9811112v1, p. 3, read clause by clause; claims checked): for every positive rational $r$ the set $\mathcal L_1(r)$ of integers $x>r^{-1}$ that cannot be the largest denominator in an Egyptian fraction representation of $r$ has zero density, and for $x\ge3$ its counting function satisfies $x\log\log x/\log x\ll_rL_1(r;x)\ll_rx\log\log x/\log x$. At $r=1$, $\mathcal L_1(1)=B$ (the integer $1$ lies in $A$), so $A$ has density $1$ and the site's order for $B$ is Martin's display (6). The site's description of $B$ is Martin's remark on p. 4 that all elements of $\mathcal L_1(r)$ are tiny multiples of prime powers ("the only ambiguity being the exact meaning of 'tiny'"), made precise in the proof (pp. 24--25): for large $x$, every $n\le x$ with a prime factor exceeding $Cx/\log x$ lies in $\mathcal L_1(1)$, and every element of $\mathcal L_1(1)$ below $x$ is at most $x/\log x$ or has a prime-power factor exceeding $x\log^{-24}x$. Acceptance evidence: the paper is published in Acta Arithmetica 95 (2000), no. 3, 231--260, a refereed journal (the arXiv listing's journal reference and the Crossref record for DOI 10.4064/aa-95-3-231-260, both), and the site accepts it. Proof coverage: the proof of Theorem 4 (pp. 24--25) was read for structure and is sketched on the theorem page; its inputs (Lemmas 9, 10 and 18, the last resting on Proposition 5, which is reduced on pp. 5--6 to Propositions 7 and 8 of Sections 4 and 5) have not been compiled, which is the remaining proof-coverage obligation. The page numbers are the arXiv preprint's, of which the library holds no file; the journal text was not compared.

Elementary facts of the commentary (verified in this paragraph). Closure under multiplication: if $1=\sum_i1/m_i$ has largest denominator $n$ and $1=\sum_j1/m'_j$ has largest denominator $n'>1$, replace the term $1/n$ by $\sum_j1/(nm'_j)$; the new denominators $nm'_j\ge2n$ exceed every other $m_i$ and are distinct, and the largest is $nn'$. Doubling: for $n>1$, from $\sum1/m_i=1$ (all $m_i\ge2$) one gets $\frac12+\sum1/(2m_i)=1$ with distinct denominators $2<2m_1<\cdots<2n$. No prime power: if $n=p^k$ were the largest denominator, the other reciprocals would sum to $1-1/p^k$, whose lowest-terms denominator is $p^k$, while every other denominator is below $p^k$ and so has $p$-adic valuation below $k$. The monograph's sharper exclusion $x_n\ne ap^k$ for $p>a!\log a$ is not verified on this page.

Explore $A$: the finer questions. Martin's Theorem 3 (p. 3; claims checked) shows that for each $j\ge2$ only finitely many integers cannot be the $j$th-largest denominator of a representation of $1$, and none once $j$ is large; he suggests (p. 3, unproved) that $\{2,4\}$ may be the full list for $j=2$ and that every $j\ge3$ may exclude nothing. OEIS A092671 records a conjectured characterization of $A$ (verified to $2\cdot10^5$ by its contributors) in terms of the largest prime-power divisor; it is a data observation, not a theorem.

Search scope. The site's problem, discussion and proof-claim pages; the community database record; the formal-conjectures directory; the arXiv listing for math/9811112 (one version; journal reference as above); the Crossref record of the article; the Semantic Scholar citation list of the paper (nine records, none on the density of $A$); an arXiv API search for abstracts naming Egyptian fractions and the largest denominator (two records, both Martin's); OEIS A092671; the primary sources [Ma00] and [ErGr80] read as stated. Not searched: MathSciNet, zbMATH, Google Scholar, X. Nothing found changes the status.

Remaining gaps. (1) Martin's proof is compiled as a statement with a structural sketch; Lemmas 9, 10 and 18 and Sections 3--5 are not compiled. (2) The journal version is not held. (3) The exact sets $\mathcal L_2(1)$ and $\mathcal L_3(1)$ and the A092671 characterization are open data questions, not part of the status. (4) The formal-conjectures statement and the external Lean proof it tags are not built or audited by this corpus.

Progress and known results

Martin's Theorem 4: $B=\mathbb N\setminus A$ has counting function $\asymp x\log\log x/\log x$, so $A$ has density $1$; its elements are the tiny multiples of prime powers in the sense of the proof. Martin's Theorem 3: the analogous exceptional sets for the second-largest and later positions are finite and eventually empty. The companion asymptotic for the least possible largest denominator is Problem 285 (Martin's Theorem 2), and the count of representations of $1$ with denominators at most $N$ is Problem 297.

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
• martin_2000_denser_egyptian_fractions
• martin_2000_denser_egyptian_fractions / theorem_2
• martin_2000_denser_egyptian_fractions / theorem_3
• martin_2000_denser_egyptian_fractions / theorem_4
• martin_shi_2021_algorithm_egyptian_fraction_representations_restricted_denominators / conjecture_4_1