Problem 295

../

---

Statement. Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with

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

Is it true that

$$\lim_{N\to \infty} k(N)-(e-1)N=\infty?$$

Formulation. $k(N)$ is the quantity $U_N$ of Monthly problem E2232 (1970–1971): the least number of distinct unit fractions summing to $1$ whose largest term is at most $1/N$, so that every denominator is at least $N$. The main term $(e-1)N$ is settled; the question is whether the excess $k(N)-(e-1)N$ tends to infinity, that is, whether its limit inferior is infinite; an unbounded excess (an infinite limit superior) would not answer it. Small values from the Monthly problem: $k(1)=1$, $k(2)=3$, $k(3)=5$ (since $1=\frac13+\frac14+\frac15+\frac16+\frac1{20}$).

Status. Open: the site's label is OPEN. The Erdős–Straus solution of 1971 proves $(e-1)N-c_2<k(N)<(e-1)N+c_1N/\log N$ and says the divergence "seems to us certain" but is unproved; the 1980 monograph repeats both. No proof or disproof of the divergence was found in the search whose scope the Current assessment records; the one proof claim on the site is partial, AI-assisted and unreviewed. This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/295, accessed 2026-09-17: the problem page (OPEN; last edited 1 October 2025), its empty discussion thread and its proof-claim tab with one partial claim. Cite as: T. F. Bloom, Erdős Problem #295, https://www.erdosproblems.com/295, 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.
• [ErSt71b] Ruderman, H. D. (proposer), Erdős, P. and Straus, E. G. (solvers), Problem E2232, Representation of 1 by Egyptian fractions. Amer. Math. Monthly 78 (1971), no. 3, 302--303, doi:10.2307/2317539.
• [Er50c] Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210; Theorem 1, p. 195, the input of the upper bound.
• [Cr01] Croot, E. S., III, On unit fractions with denominators in short intervals. arXiv:math/9904181 (1999); Acta Arith. 99 (2001), no. 2, 99--114, doi:10.4064/aa99-2-1. Context.
• [Ma99] Martin, G., Dense Egyptian fractions. arXiv:math/9804045 (1998); Trans. Amer. Math. Soc. 351 (1999), no. 9, 3641--3657, doi:10.1090/S0002-9947-99-02327-2. Context.
• OEIS A192881, linked by the site.

Formalization. Statement only. The file ErdosProblems/295.lean of formal-conjectures, at the revision linked (fetched 2026-09-17), defines k N by Nat.find on a helper lemma exists_k whose proof is sorry, declares erdos_295 : answer(sorry) ↔ Filter.atTop.Tendsto (fun N => k N - (rexp 1 - 1)*N) Filter.atTop under category research open, and records the Erdős–Straus bounds as the variant erdos_295.variants.erdos_straus under research solved, also with proof sorry. The formal $k(N)$ agrees with the site's for every $N\ge1$. The file is a statement, not a proof; no build or audit of it is recorded. The community database (fetched 2026-09-17) records a formalized statement and no formal proof.

Current assessment

The question. The site states the problem as above, shows OPEN, cites [ErGr80] and [ErSt71b], and records the bounds $-c<k(N)-(e-1)N\ll N/\log N$. Printed p. 35 of the monograph (Erdős–Graham 1980) defines $U_n=\min\{k:\{x_1,\ldots,x_k\}\in\mathcal X,\ x_1\ge n\}$, says "It seems likely that $\lim_n\{U_n-(e-1)n\}=\infty$ but we cannot prove this", and quotes the Erdős–Straus bounds. The exact question is the divergence of the excess.

Known results.

• Erdős–Straus, inequality (1) (1971): $(e-1)n-c_2<U_n<(e-1)n+c_1n/\log n$. The lower bound compares the harmonic sum with a logarithm. The upper bound takes the consecutive denominators $n,n+1,\ldots,m$ with $m=en+O(1)$ whose reciprocals fall just short of $1$, and represents the deficit $u/v$, which is below $1/(m+1)$ and has $v<e^{2m}$, by fewer than $c\log v/\log\log v$ further unit fractions using Erdős's Theorem 1 of 1950; their denominators exceed $m+1$ automatically, and they cost $O(n/\log n)$ terms. The published solution is on JSTOR (pp. 302–303); the library holds no file of it. The solvers add that the divergence "seems to us certain" but is unproved; the editorial note after the solution records that several solvers conjecture $U_n\le2n$.
• Adjacent results that are not the question. Croot's short-intervals theorem (1999 preprint; Acta Arith. 2001): for every rational $r>0$ and every $N>1$ there is a representation $r=\sum1/x_i$ with $N<x_1<\cdots<x_k\le(e^r+O_r(\log\log N/\log N))N$, with best-possible error term. With $r=1$ the representation's denominators are distinct integers in $(N,(e+O(\log\log N/\log N))N]$, so it has at most $(e-1)N+O(N\log\log N/\log N)$ terms. The theorem therefore gives only $k(N)\le(e-1)N+O(N\log\log N/\log N)$, weaker than the Erdős–Straus upper bound, and no lower bound on the excess, so it neither proves nor refutes the divergence. Martin's Theorem 1 (1998 preprint; Trans. Amer. Math. Soc. 1999) is the opposite extreme: for large $x$ a representation of $r$ using more than $(C(r)-\eta)x$ of the integers up to $x$, with $C(r)=(1-\log2)(1-e^{-r/(1-\log2)})$. The density and coloring theorems of Bloom and of Croot (2003) produce unit subsums inside dense or smooth sets and do not address the least number of terms above a threshold; Liu and Sawhney's reciprocal-mass threshold is likewise a different quantity, as their card records. Croot's Mathematika paper of 1999 (card) concerns which integers are sums of unit fractions with denominators at most $x$ (Problem 308 and Problem 309), not $k(N)$.

Finite values (leads). OEIS A192881, "Number of terms for the shortest Egyptian fraction representation of 1 starting with $1/n$", lists $1,3,5,8,10,11,13,15,17,19,21,23,25,26,28,30$ for $n=1,\ldots,16$. It fixes $n_1=n$, whereas $k(N)$ allows $n_1>N$, so each listed value is an upper bound for $k(n)$; the entry's example for $n=3$ matches $k(3)=5$. It is a community record, not a refereed source. An AI-assisted public report of 11 August 2026 (erdosproblemaday.com/day/295-k17-exact) claims $k(17)=32$ and $33\le k(18)\le35$ by exhaustive search with certificates, labels itself partial, and says it does not touch the asymptotic question; no check of it is recorded.

The site's proof claim (lead). The proof-claim tab lists one partial claim, submitted on 4 September 2026 by the user tienxion and disclosed as produced with substantial help from an AI model (GPT-6 through Codex); it links a PDF in that user's GitHub repository. Its summary: for each fixed $B>1$, an explicit lower bound of order $N/\log N$ for the excess when all denominators are at most $N^{B+o(1)}$, and the statement that bounded excess forces exponential denominators. It does not claim the unrestricted problem. The site says that appearing there is no guarantee of correctness; no check of the claim is recorded. It settles no instance of the question, since it bounds a quantity restricted to polynomially bounded denominators, so it has no claim page and does not affect the status.

Search scope. Routes; none found a proof or disproof of the divergence.

• The site's three pages; the community database (open; formalized statement; no proof URL); formal-conjectures 295.lean at the pinned commit (statement only).
• arXiv: the abstract pages of math/9904181 (one version) and math/9804045 (one version, "to appear in Trans. Amer. Math. Soc."); API metadata searches for "Egyptian fractions" AND "number of terms" (two records: Martin's 1998 sequel on denser Egyptian fractions and a 2020 paper on harmonic sums), "unit fractions" AND denominators AND distinct (eleven) and a sweep of 2025–2026 abstracts mentioning "Egyptian fractions" or "unit fractions" (29 records); none on $k(N)$.
• Crossref: the Acta Arithmetica and Transactions records. Semantic Scholar: the Monthly solution's DOI is not indexed. OEIS A192881.
• The primary sources: [ErSt71b]; [Er50c] Theorem 1; the arXiv preprints of Croot 1999 and Martin 1998; [ErGr80] p. 35.

Not searched: MathSciNet, Google Scholar, X.

Proof coverage. Inequality (1) is paged with its short proof (claims checked; the argument was read in full, not independently reviewed). Nothing else is compiled: no source proves or disproves the statement.

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
• bloom_2021_density_conjecture_about_unit_fractions
• croot_1999_unit_fractions_denominators_short_intervals
• croot_1999_unit_fractions_denominators_short_intervals / main_theorem
• croot_2003_coloring_conjecture_about_unit_fractions
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / theorem_1
• martin_1998_dense_egyptian_fractions
• martin_1998_dense_egyptian_fractions / theorem_1
• ruderman_1971_e2232_representation_1_egyptian_fractions_problem
• ruderman_1971_e2232_representation_1_egyptian_fractions_problem / conjecture_p303
• ruderman_1971_e2232_representation_1_egyptian_fractions_problem / inequality_1