Problem 294

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claims/: The 1 claim page of Problem 294, one per claimant's result; the problem's standing derives from them.

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Statement. Let $N\geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to

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

with $t=n_1<\cdots <n_k\leq N$. Estimate $t(N)$.

Formulation. The site's wording as read on 2026-09-17 (page last edited 18 November 2025). The denominators are distinct integers in $[t,N]$ and the number of terms is free. Since $t=1$ always admits the one-term representation, $t(N)\ge2$; for $N\ge2$ no representation starts at $t=N$, so $t(N)\le N$. In the Erdős–Graham monograph the same quantity is written $k_1(n)$.

Status. PROVED, the site's label, which attaches to the estimate: Liu and Sawhney (Int. Math. Res. Not. 2026) determine $t(N)$ up to a factor $(\log\log N)^3(\log\log\log N)^{O(1)}$, namely $N/((\log N)(\log\log N)^3(\log\log\log N)^{O(1)})\ll t(N)\ll N/\log N$, the upper bound being the one Erdős and Graham had stated. This is the accepted claim Liu and Sawhney 2024, refereed and credited by the site's curator. The derived standing is solved, answered rather than proved, because the question asks for an estimate rather than a yes or no. The exact order of $t(N)$ inside that window is not known.

Source. erdosproblems.com/294, accessed 2026-09-17: the problem page (PROVED; last edited 18 November 2025), its empty discussion thread and its empty proof-claim tab. The site cites [ErGr80, p. 35] as the problem's source and [LiSa24] in its commentary. Cite as: T. F. Bloom, Erdős Problem #294, https://www.erdosproblems.com/294, 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), printed p. 35. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [LiSa24] Liu, Y. P. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113v1 (10 April 2024); Int. Math. Res. Not. 2026, no. 2, rnaf382, DOI 10.1093/imrn/rnaf382, published online 14 January 2026. Theorem 1.6. Library home: liu_2024_further_questions_regarding_unit_fractions.

Formalization. No formal-conjectures statement: formal-conjectures has no ErdosProblems/294.lean, and the community database records the problem as unformalized. The external collection plby/lean-proofs holds, since 17 August 2026, a file declaring itself a formalization of a solution, with Liu and Sawhney as informal authors and Codex and GPT-5.6 Sol as formal authors; it is a formalization link on the claim page, where its statement is described, and nothing was built or audited by this corpus.

Current assessment

The question. On 2026-09-17 the site asks for the order of $t(N)$ and shows PROVED. Its commentary credits Erdős and Graham with the upper bound $t(N)\ll N/\log N$ and with no knowledge of the true size, and records Liu and Sawhney's two-sided estimate, which it describes as a solution up to a factor $(\log\log N)^{O(1)}$:

$$\frac{N}{(\log N)(\log\log N)^3(\log\log\log N)^{O(1)}}\ll t(N)\ll\frac{N}{\log N}.$$

The thread and the proof-claim tab are empty. The community database record (teorth/erdosproblems) says proved, unformalized.

Status support. The status-defining source is Liu and Sawhney's Theorem 1.6, arXiv:2404.07113v1, p. 3, read clause by clause on the page image: with $t(N)$ defined exactly as on the site, the two-sided bound displayed above holds. The paper appeared in Int. Math. Res. Not. 2026, no. 2, rnaf382 (received 28 October 2025, accepted 23 December 2025, online 14 January 2026, per the publisher's record read), which is the acceptance evidence; the library retains arXiv v1, and the published text has not been compared, so all locators are v1 locators. The proof (pp. 13--14) was read for its structure: the upper bound takes a prime $t>10N/\log N$ and shows that a representation starting at $1/t$ would force a congruence modulo $t$ that the small least common multiple of the other quotients cannot satisfy; the lower bound represents $1$ starting from $1/t$ by one application of the paper's Lemma 4.1 for $t\le N^{9/10}$, and by two, at two scales, for larger $t$; the lemma produces prescribed smooth fractions from denominators in an interval of constant ratio by way of its Proposition 3.2. The theorem page records that a complete rewrite of both bounds and their dependencies is not compiled; the library's proof coverage for this theorem is statement and sketch only.

Two remarks on the source. On p. 3 the authors report that they knew of no earlier study of this question, and they label it "Problem # 305" in the site's numbering; the statement matches this problem, and the site's Problem 305 is the question on the largest denominator of a representation of $a/b$. The site's attribution of the upper bound to Erdős and Graham is borne out by printed p. 35 of the monograph, read on the page image: with $k_r(n)$ the least integer not occurring as $x_r$ in any $\{x_1<\cdots<x_t\}$ with $x_t\le n$ and reciprocal sum one, they write that $k_1(n)<cn\log\log n/\log n$ is easy to show and that "with a slight refinement" $k_1(n)<cn/\log n$, adding "We have no idea of the true value of $k_r(n)$ or even of $k_1(n)$"; no proof is given there, and Liu and Sawhney prove the upper bound themselves.

What remains open. The order of $t(N)$ between $N/((\log N)(\log\log N)^{3+o(1)})$ and $N/\log N$; nothing found narrows the window.

Search scope. The site's problem, discussion and proof-claim pages as read; the community database record; the formal-conjectures directory listing; the arXiv listing for 2404.07113; the Oxford Academic article record; the Semantic Scholar citing-paper records for the paper (five records: arXiv:2607.04157, 2502.02200, 2406.07218, 2404.16016 and a formalization paper, none about $t(N)$); the arXiv API listing of the sixty most recent abstracts mentioning unit or Egyptian fractions (to 7 September 2026); and two general web searches. Not searched: MathSciNet, zbMATH, full-text search engines for scholarly literature, X. No later work on $t(N)$ was found; this is a bounded negative finding.

Remaining gaps. The proof of Theorem 1.6 is not compiled beyond a sketch; the published text is uncompared; the monograph's "slight refinement" is unrecorded; the external Lean file is not built or audited by this corpus, and no formal-conjectures statement exists.

Progress and known results

• Erdős and Graham (1980, printed p. 35): $t(N)\ll N/\log N$, stated without proof, adding that the true order was unknown to them.
• Liu and Sawhney's Theorem 1.6 (2024; published 2026): the two-sided estimate above, proved by the method of their Theorem 1.5 on the largest denominator of a representation of $a/b$, the subject of Problem 305.

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_1979_old_new_problems_results_combinatorial_number
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• liu_2024_further_questions_regarding_unit_fractions
• liu_2024_further_questions_regarding_unit_fractions / theorem_1_6