Problem 148

../

claims/: The 3 claim pages of Problem 148, one per claimant's result; the problem's standing derives from them.

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Statement. Let $F(k)$ be the number of solutions to

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

where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$.

Formulation. A solution is a set of $k$ distinct positive integers whose reciprocals sum to $1$, counted once; $n_1=1$ occurs only for $k=1$, so $F(1)=1$, $F(2)=0$ and $F(3)=1$ (the set $\{2,3,6\}$). The question is about the growth of $F(k)$ as $k\to\infty$. It counts representations of a fixed length $k$; the number of subsets of $\{1,\ldots,N\}$ with reciprocal sum $1$, a count with a denominator cutoff and no length restriction, is Problem 297 and is not $F(k)$.

Status. Open on the site: the label is OPEN (page last edited 27 September 2025; no proof claim on its tab). The order of the double logarithm is fixed: $ck\le\log\log F(k)\le Ck$ for all large $k$, by Corollary 1.2 of the OpenAI mathematics release's manuscript of 25 September 2026 (its claim page); the lower half is new and replaces the published $\exp(\exp(ck/\log k))$, the upper half was known from explicit bounds of the form $c^{2^k}$, the best of them due to Browning and Elsholtz and to Elsholtz and Planitzer (see Upper bound below). No asymptotic formula, no estimate up to constant factors and no value of the constant in $\log\log F(k)$ is known.

Source. erdosproblems.com/148, accessed 2026-09-17: the problem page (OPEN; last edited 27 September 2025; source keys [ElPl21], [ErGr80], [Ko14]), its two-comment discussion thread and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #148, https://www.erdosproblems.com/148, 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. 32.
• [Ko14] Konyagin, S. V., Double exponential lower bound for the number of representations of unity by Egyptian fractions. Math. Notes 95 (2014), 277--281, doi:10.1134/S0001434614010295; Russian original Mat. Zametki 95 (2014), no. 2, 312--316, doi:10.4213/mzm10417.
• [ElPl21] Elsholtz, Christian and Planitzer, Stefan, Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. 53 (2021), no. 3, 695--709, doi:10.1112/blms.12452; arXiv:2012.05984v1 (2020).
• [El16] Elsholtz, C., Egyptian fractions with odd denominators. Q. J. Math. 67 (2016), no. 3, 425--430, doi:10.1093/qmath/haw020; arXiv:1606.02117v1 (2016).
• [DoTa25] 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 2026, doi:10.1017/S0305004126102102. Context: the link between $F(k)$ and the smallest missing denominator of Problem 293.
• [CFHMPSV25] Conlon, D., Fox, J., He, X., Mubayi, D., Pham, H. T., Suk, A. and Verstraëte, J., A question of Erdős and Graham on Egyptian fractions. Discrete Analysis 2025:28, doi:10.19086/da.154329. Context: a count with a different parameter.
• [Er50] Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210, p. 194. Context: the earliest statement of the counting question that the search located.

Formalization. Statement only. The file ErdosProblems/148.lean of formal-conjectures, added on 27 September 2026 and on main on 2026-10-07 identical to the linked revision, defines F k as the ncard of the finite sets of positive integers of size $k$ with reciprocal sum $1$ and declares erdos_148 : (fun k ↦ (F k : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) under category research open, with proof sorry; its two research solved variants record Konyagin's lower bound with the constant $(\log2)(\log3)/3$ and Elsholtz–Planitzer's upper bound with the exponent $(1/5+o(1))2^k$ of the squared Vardi constant, both also sorry. Apart from two proved test lemmas (F_one, $F(1)=1$, and u_first_values), no declaration in the file has a proof, and nothing in it bears on the release's result. The file is not built here. The community database records the statement as formalized, with no formal-proof URL (export of 2026-10-06; the export of 2026-09-17 preceded the file). The release's formalization of Corollary 1.2, built and audited by the corpus's verification, is recorded on the claim page linked under Status.

Current assessment

The question. On 2026-09-17 the site asks for good estimates of $F(k)$, cites [ErGr80, p. 32], shows OPEN and marks the problem as not resolvable by a finite computation, lists no proof exposition and no proof claim, and records in its commentary a lower bound of the shape $2^{c^{k/\log k}}$ with an absolute $c>0$, attributed to Konyagin [Ko14], and an upper bound written as $c_0^{(1/5+o(1))2^k}$ with $c_0=1.26408\cdots$, the constant it calls the Vardi constant, attributed to Elsholtz and Planitzer [ElPl21]. The form $2^{c^{k/\log k}}$ is $\exp(\exp((\log c)\,k/\log k+\log\log2))$, the shape of Konyagin's bound; the upper bound's constant is discussed below.

Claims. Three claim pages, each accepted, partial and proved: [[problems/unit_fractions/E0148/claims/2014_01_01_konyagin|Konyagin's lower bound]] and [[problems/unit_fractions/E0148/claims/2020_12_10_elsholtz_planitzer|Elsholtz and Planitzer's upper bound]], both on their refereed publication (see Lower bound and Upper bound below), and [[problems/unit_fractions/E0148/claims/2026_09_25_openai|the OpenAI release's Corollary 1.2 (25 September 2026)]], on formalized evidence: $ck\le\log\log F(k)\le Ck$ for all $k\ge k_0$, the constants not made explicit. That 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 half replaces the lower bounds below for large $k$; its upper half adds nothing to Elsholtz–Planitzer.

Origin. Printed p. 32 of the 1980 monograph defines $\mathcal X_n$ as the sets $\{x_1,\ldots,x_n\}$ with $\sum1/x_k=1$ and $0<x_1<\cdots<x_n$, says "it would be interesting to have asymptotic formulas or even good inequalities for $|\mathcal X_n|$", and records: "The only estimates currently known are due to Straus and the authors. These are $e^{n^{2-\varepsilon}}<|\mathcal X_n|\le c_0^{2^{n+1}}$ where $c_0=\lim u_n^{1/2^n}=1.264085\ldots$ (see [Ah-Sl (73)]). Perhaps the lower bound can be replaced by $c_0^{2^{n(1-\varepsilon)}}$." Its $u_n$ ($u_1=1$, $u_{n+1}=u_n(u_n+1)$, printed p. 30) is $1,2,6,42,\ldots$, one less than Sylvester's sequence $2,3,7,43,\ldots$, so this $c_0$ is the Vardi constant $E=1.264084\ldots$ ([[../library/number_theory/erdos_1980_old_new_problems_results_combinatorial_number_theory/_index|monograph card]]). Erdős had already listed the count in 1950 among some interesting, as yet unsolved problems concerning the solutions of the equation: for given $n$ he asks for the number $f_1(n)$ of positive integer solutions and the number $f_2(n)$ of solutions with increasing denominators, or for functions asymptotically equal to them ([[../library/unit_fractions/erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine/conjectures_p194|Erdős 1950, p. 194]]).

Lower bound. [[../library/unit_fractions/konyagin_2014_double_exponential_lower_bound_number_representations/theorem_1|Konyagin's Theorem 1]] (Mat. Zametki 95:2 (2014), p. 312; $|\mathbf X_n|$ there is $F(n)$) states that, as $n\to\infty$,

$$F(n)\ \ge\ \exp\Bigl(\exp\Bigl(\Bigl(\frac{(\ln2)(\ln3)}{3}+o(1)\Bigr)\frac{n}{\ln n}\Bigr)\Bigr),$$

with the same bound for every positive rational (Corollary 1, p. 314) and the monotonicity $F(n)\le F(n+1)$ (inequality (1), p. 312, by an explicit injection). The printed proof has a known defect: the displayed identity on p. 314 immediately before its equation (4) is false (it fails at $k=2$, $m=1$). This was reported in the site's discussion thread on 5 September 2025, and a comment of 29 December 2025 reports the author's agreement, a corrected identity and a missing odd-$m$ hypothesis in his Lemma 1, with the main result unaffected; the corrected argument is unpublished and unverified. Independently of that step, [[../library/unit_fractions/elsholtz_2016_egyptian_fractions_odd_denominators/corollary_1_2|Elsholtz's Corollary 1.2]] (arXiv v1, p. 3; Q. J. Math. 67 (2016)) proves that for odd $k$ large enough the number of representations with distinct odd denominators is at least $\exp(\exp(ck/\log k))$ for some $c>0$, by a construction that does not use Konyagin's identities. Odd-denominator solutions are solutions, and Konyagin's inequality (1) carries the bound from $k-1$ to even $k$ with the constant halved, so $F(k)\ge\exp(\exp(c'k/\log k))$ for all large $k$ with an unspecified $c'>0$ (a two-line deduction of this page, not a source statement). The doubly exponential order of $F(k)$ therefore rests on a published route without the disputed step, while the constant $(\ln2)(\ln3)/3$ rests on Konyagin's printed proof. The earlier lower bounds were $e^{n^{2-\varepsilon}}$ (1980) and $\exp(c_1n^3/\ln n)$, the latter recorded in Konyagin's display (2) and attributed by Elsholtz (2016, p. 2) to Sándor (Period. Math. Hungar. 47 (2003), 215--219; not held). The release's accepted claim gives $\log\log F(k)\ge ck$ for all large $k$, which exceeds every bound of this paragraph; the constant $c$ is not explicit.

Upper bound. [[../library/unit_fractions/elsholtz_2021_sums_four_more_unit_fractions_approximate/corollary_3|Elsholtz--Planitzer's Corollary 3(2)]] (arXiv v1, p. 5; Bull. Lond. Math. Soc. 53 (2021)): with $u_0=1$, $u_{n+1}=u_n(u_n+1)$ and $c_0=\lim u_n^{2^{-n}}=1.5979102\ldots$ (their Remark 3), for every $\varepsilon>0$ and $k\ge k(\varepsilon)$,

$$F(k)\ \le\ f_k(1,1)\ <\ c_0^{(2/5+\varepsilon)2^{k-1}} \qquad\text{i.e.}\qquad F(k)<E^{(2/5+\varepsilon)2^k},\quad E=\sqrt{c_0}=1.264084\ldots,$$

where $f_k(1,1)$ counts nondecreasing $k$-tuples with reciprocal sum $1$ (repetitions allowed), of which the distinct increasing solutions are a subset. The corollary follows from their Theorem 2, the lifting of the four-fraction Theorem 1 to $f_k(m,n)\ll_\varepsilon(kn)^\varepsilon(k^{4/3}n^2/m)^{(8/5)2^{k-5}}$; the paper refers the corollary's proof to two earlier papers. The site's commentary writes this bound as $c_0^{(1/5+o(1))2^k}$ with $c_0=1.26408\ldots$, pairing the source's exponent with the monograph's normalization of the constant; as the source prints it, the exponent of the Vardi constant is $(2/5+o(1))2^k$, twice the site's, and the printed bound is then weaker than the Browning--Elsholtz bound below, which the paper improves. The successive improvements of the lifted exponent ($5/3$, $28/17$, $8/5$) give $5/24$, $7/34$ and $1/5$ as the coefficient of $2^k$ in the exponent of $E$, so the site's form is what Corollary 3(2) gives when read with $u_1=1$; the [[problems/unit_fractions/E0148/claims/2020_12_10_elsholtz_planitzer|claim page]] gives the arithmetic, and this page records both forms. The earlier upper bounds were the monograph's $E^{2^{n+1}}$ (1980) and Browning and Elsholtz's $E^{(5/3+\varepsilon)2^{k-3}}=E^{(5/24+\varepsilon/8)2^k}$ (Illinois J. Math. 55 (2011); not held, as restated in Elsholtz 2016, display (1.2), whose $c_0=1.264\ldots$ is $E$, the sequence started at $u_1=1$, and whose lower-bound constant does not match Konyagin's theorem; see the card; Elsholtz and Planitzer's 2020 paper, arXiv:1805.02945v1, p. 1, states it as $E^{(5/24+\varepsilon)2^k}$ with $u_1=1$).

The gap. $\log F(k)$ lies between $\exp(ck)$ and $(\tfrac5{24}+o(1))2^k\log E$ for large $k$ (Browning and Elsholtz's bound as the later papers restate it; $(\tfrac15+o(1))2^k\log E$ if Corollary 3(2) is read with $u_1=1$), so $\log\log F(k)/k$ is eventually between $c$ and $\log2+o(1)$; before the release the lower end was $\exp(c'k/\log k)$. The 1980 guess $c_0^{2^{n(1-\varepsilon)}}$ would put $\log F(k)$ near $2^{k(1-\varepsilon)}$, close to the upper bound, and corresponds to the slope $\log\log F(k)/k$ tending to $\log2$, which is open. Elsholtz--Planitzer's Conjecture 1 (for fixed $k$ and $m$, $f_k(m,n)\ll\exp(C_{m,k}\log n/\log\log n)$ as $n\to\infty$) concerns the dependence on $n$ and says nothing about $F(k)$. The search recorded below found no source that improved either side or gave an asymptotic formula; the release's Corollary 1.2 of 25 September 2026 improves the lower side.

Counts that are not $F(k)$.

• The smallest denominator missing from every $k$-term representation, $v(k)$ of Problem 293, satisfies $v(k)\le|D_k|+2\le kF(k)+2$ ([[../library/unit_fractions/doorn_2025_smallest_denominator_not_contained_unit_fraction/inequality_1_2|van Doorn--Tang, inequality (1.2)]], arXiv v2 p. 1; the paper is published online in Math. Proc. Cambridge Philos. Soc., 2026). Upper bounds for $F(k)$ thus bound $v(k)$; they write the result as $c_0^{(1/5+o(1))2^k}$ with $c_0=1.264085\ldots$, the site's form, citing Corollary 3, so the normalization remark above applies to it too. Their lower bound $v(k)\ge e^{ck^2}$ gives through (1.2) only $F(k)\ge(e^{ck^2}-2)/k$, far below the doubly exponential lower bounds above.
• Conlon and collaborators' Theorem 1 counts subsets of $\{1,\ldots,n\}$ with reciprocal sum $1$, giving $2^{c_1n+o(n)}$ with $c_1\approx0.91117$: the parameter is a denominator cutoff, not a length, and no bound for $F(k)$ is drawn from it.
• Elsholtz's odd-denominator count and Erdős's 1950 $f_1(n)$ (positive integer solutions with repetitions allowed; Elsholtz--Planitzer's $f_k(1,1)$ in nondecreasing order) are restricted or enlarged variants used above only as bounds.

Unverified web items (none is progress). On 2026-09-17 the discussion thread held two comments, which the site does not verify: one of 5 September 2025 (Quanyu Tang) reporting the false identity in Konyagin's proof and noting that Elsholtz's 2016 result covers the order of the lower bound, and one of 29 December 2025 (posted as "Woett") relaying by e-mail the author's corrected identity and the odd-$m$ condition. The correction is recorded as reported and is unverified. A third comment, of 22 September 2026, reports the exact values $F(1),\ldots,F(8)=1,0,1,6,72,2320,245765, 151182379$ from an exhaustive search, with its code on GitLab, and says that it is not a theoretical result; the values agree with $F(3)=1$ above and with OEIS A006585, which the site links together with A076393. A computation of finitely many values settles nothing in an estimate question, so it has no claim page.

Search scope. The status rests on these dated routes; none had found a proof of an asymptotic formula, an improved bound or a proof claim. The release's manuscript of 25 September 2026, recorded under Claims, postdates this search.

• The site: problem page, discussion thread, proof-claim tab; the community database record (open, unformalized); formal-conjectures as fetched on 2026-09-17 (no file 148).
• The primary sources cited above: [Ko14] (Russian original, printed pp. 312--314), [ElPl21] (arXiv v1, pp. 1--5), [El16] (arXiv v1, pp. 1--3), [ErGr80] (p. 32), [DoTa25] (arXiv v2, pp. 1--2), [Er50] (p. 194).
• Publication records: arXiv abstract pages of 2012.05984 (v1 only, no journal reference listed), 1606.02117 (v1 only), 2512.22083 (v1, v2, MPCPS reference); Crossref records for doi:10.1112/blms.12452, doi:10.1093/qmath/haw020 and doi:10.1134/S0001434614010295; the MathNet record of [Ko14] (no erratum listed).
• arXiv API metadata searches: abs:"Egyptian fractions" AND abs:"number of" AND (abs:representations OR abs:solutions) (eight records, none newer than 2025 on this count) and abs:"unit fractions" AND (abs:"number of solutions" OR abs:"number of representations" OR abs:"representations of 1" OR abs:"decompositions of 1") (six records; the newest relevant is [ElPl21]). The API searches titles and abstracts only, so these zeros are weak.
• A general web search engine: queries for a correction or erratum to [Ko14] (none found) and for 2026 work on the number of $k$-term representations of $1$ (nothing beyond the sources above).

Not searched: MathSciNet, zbMATH, Google Scholar full text, X. Not held: Browning--Elsholtz 2011, Sándor 2003, the journal versions of [ElPl21] and [El16], the English translation of [Ko14].

Proof coverage. Nothing establishes a status other than open, so there is no resolving proof to compile. 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 recorded for structure only on the source card. The two published bounds, accepted on their refereed publication, are recorded at statement level on their result pages; Konyagin's printed proof has the known false step, and the proof of Elsholtz--Planitzer's corollary is delegated to earlier papers not held. No proof has been rewritten or independently reviewed.

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
• conlon_2024_question_erdos_graham_egyptian_fractions
• doorn_2025_smallest_denominator_not_contained_unit_fraction
• doorn_2025_smallest_denominator_not_contained_unit_fraction / inequality_1_2
• elsholtz_2016_egyptian_fractions_odd_denominators
• elsholtz_2016_egyptian_fractions_odd_denominators / corollary_1_2
• elsholtz_2016_egyptian_fractions_odd_denominators / theorem_1_1
• elsholtz_2020_number_solutions_erdos_straus_equation
• elsholtz_2020_number_solutions_erdos_straus_equation / corollary_3
• elsholtz_2020_number_solutions_erdos_straus_equation / theorem_2
• elsholtz_2021_sums_four_more_unit_fractions_approximate
• elsholtz_2021_sums_four_more_unit_fractions_approximate / corollary_3
• elsholtz_2021_sums_four_more_unit_fractions_approximate / theorem_1
• elsholtz_2021_sums_four_more_unit_fractions_approximate / theorem_2
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / conjectures_p194
• konyagin_2014_double_exponential_lower_bound_number_representations
• konyagin_2014_double_exponential_lower_bound_number_representations / theorem_1
• openai_2026_short_egyptian_fractions
• openai_2026_short_egyptian_fractions / corollary_1_2
• openai_2026_short_egyptian_fractions / theorem_1_1