Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 297

claims/: The 3 claim pages of Problem 297, one per claimant's result; the problem's standing derives from them.

Statement. Let $N\geq 1$ . How many $A\subseteq \{1,\ldots,N\}$ are there such that $\sum_{n\in A}\frac{1}{n}=1$ ?

Status. Solved, in the site's label, at the level of the exact exponential growth rate. The answer below does not give a finite- $N$ formula or a multiplicative asymptotic.

Source. T. F. Bloom, Erdős Problem #297, accessed 2026-09-05. The site records SOLVED, shows no formalized statement and lists no proof claims.

References.

P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), printed pp. 32 and 36.
[CFHMPSV24] D. Conlon, J. Fox, X. He, D. Mubayi, H. T. Pham, A. Suk and J. Verstraëte, A question of Erdős and Graham on Egyptian fractions, Discrete Analysis 2025:28. The printed DOI 10.19086/da.154329 resolved on 2026-10-07 to a different article (Magnitude function determines generic finite metric spaces), so the journal's own page is linked.
[LiSa24] Y. P. Liu and M. Sawhney, On Further Questions Regarding Unit Fractions, IMRN 2026(2), rnaf382, DOI 10.1093/imrn/rnaf382; available arXiv v1.
[St24] S. Steinerberger, On a Problem Involving Unit Fractions, arXiv:2403.17041v5 (2024).

Formalization. The site shows no formalized statement, and formal-conjectures has no file for Problem 297. Boris Alexeev's lean-proofs collection holds a Lean 4 file that declares itself a formalization of Liu and Sawhney's solution, with Codex and GPT-5.6 Sol as formal authors; it is linked at its pinned commit on Liu and Sawhney's claim page and described under Current assessment. The corpus has not built it.

Current assessment

Claims. Two accepted full claims settle the problem at the level of the exponential rate, each refereed and each credited by the site's curator independently of its authors: Conlon and collaborators' Theorem 1 (Discrete Analysis 2025:28) and Liu and Sawhney's Theorem 1.2 (Int. Math. Res. Not. 2026). Steinerberger's earlier eventual upper bound $2^{0.93N}$ is the accepted partial claim Steinerberger's bound; its value is disproved, since it answers no to the monograph's question whether there are $2^{N-o(N)}$ such subsets, while the full claims are answered, answering yes to its question whether there are $2^{cN}$ and no to the other. The frontmatter standing is derived from these pages. The 2017 MathOverflow discussion of the relaxed count is recorded under Earlier relaxed counts below and has no claim page: it is a thread, not a dated manuscript.

The Conlon source supplies a complete ordinary proof chain relative to its explicit external inputs. Its entropy upper bound first counts subsets whose reciprocal sum is at most the target. For the exact lower bound, the absorption construction combines many partial representations with a small reserved set of denominators. The modular subset-sum estimate, successive prime-power removal and final completion explain how it reaches the target exactly while retaining the exponential rate. The source pages disclose the compilation's repairs and their sufficient application scopes.

A Lean 4 formalization of the result exists outside the corpus: the file src/latest/ErdosProblems/Erdos297.lean of Boris Alexeev's lean-proofs collection, posted 17 August 2026 and linked at its pinned commit on Liu and Sawhney's claim page, declares itself a formalization of a solution to Problem 297, names Liu and Sawhney as informal authors and Codex and GPT-5.6 Sol as formal authors, and proves that the count is $\exp((\gamma_*+o(1))N)$ , together with the base-two form of the rate and the bound $c_1<1$ . The corpus has not built or audited it, so it gives no formalized evidence. The site shows no formalized statement, the community database records the problem as informally solved and unformalized, and formal-conjectures has no file for Problem 297.

A search of the primary literature located the published Conlon article, Liu and Sawhney's publication record and Steinerberger's archived note, and no later result changing the exponential rate. Finer asymptotics are not classified here.

Neither route has been independently verified in this corpus.

Answer and conventions

Write $F(N)$ for the number in the statement. Denominators within each representation are positive and distinct; permutations do not give new representations. The original Erdős–Graham book's definitions on printed page 32 and question on page 36 use these conventions. Their instruction to drop disjointness concerns different counted representations, not repetitions inside one representation.

The published Theorem 1 of Conlon and collaborators gives

F(N)=2^{c_1N+o(N)},\qquad \lim_{N\to\infty}\frac{\log_2 F(N)}N=c_1\in(0,1).

Let $\lambda>0$ be the unique solution of

\int_0^1\frac{dy}{y(1+e^{\lambda/y})}=1.

With $p(y)=(1+e^{\lambda/y})^{-1}$ and binary entropy $h_2(t)=-t\log_2t-(1-t)\log_2(1-t)$ , the constant is

c_1=\int_0^1h_2(p(y))\,dy.

The entropy-exponent page proves the defining multiplier's existence and uniqueness and the exponent's bounds. The source reports $c_1\approx0.91117$ ; these decimal digits have not been certified here. In particular, the answer rules out $F(N)=2^{N-o(N)}$ . It does not assert $F(N)/2^{c_1N}\to1$ or identify a polynomial prefactor.

More generally, the same theorem gives $2^{c_xN+o_x(N)}$ representations of every fixed positive rational $x$ . Fix $x$ before letting $N$ grow; this is not a uniform assertion for arbitrary varying targets.

Proof methods and connections

Liu–Sawhney's Theorem 1.2 gives an independent Fourier method for the exact target one. It selects a Bernoulli subset of smooth denominators, establishes enough probability of an exact hit, and converts that probability to a count using exponential weights. Its natural-log constant is

\gamma=\lambda+\int_0^1\log(1+e^{-\lambda/y})\,dy =c_1\log2.

The entropy identity identifies the constants, not the two lower-bound constructions. The counting theorem is distinct from the same paper's density theorem concerning Problem 298 and Problem 299. The arXiv v1 and the published full text remain distinct evidence; their full proofs have not been compared.

The site also cross-references Problem 362.

Earlier relaxed counts and source history

The 2017 MathOverflow discussion, opened by Mikhail Tikhomirov on 14 September, concerns subsets whose reciprocal sum is at most one. Lucia's accepted answer and other contributors discuss the sharp exponential rate through entropy and exponential tilting. The relaxed upper bound applies to the exact-sum family; the relaxed lower bound does not supply the exact-sum lower bound. The acceptance is a public community record, not a journal or formal verification record. Conlon and collaborators' published article and Steinerberger's archival note acknowledge that discussion and the later exact-sum work.

Steinerberger's 2024 note proves the earlier bound $2^{0.93N}$ for the larger relaxed family, for sufficiently large $N$ . Its arXiv v5, dated 28 April 2024, says that the note was kept for archival purposes and was not submitted to a journal. Its discussion of the then interesting exact-sum question is historical background.

The Conlon version record distinguishes the eight-page April 2024 v1 from the published thirteen-page article, which is byte-identical to the December 2025 arXiv v2. The published article appeared on 19 December 2025 in Discrete Analysis. Liu–Sawhney appeared online in International Mathematics Research Notices on 14 January 2026; the arXiv record lists one version, v1 of 10 April 2024. Publication metadata does not establish that every formula in that v1 agrees with the published text.

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.

Linked from (43)

Problem 148 Problem 285 Problem 292 problems/unit_fractions/E0297/claims Problem 300 The largest unit-subsum-free subset has size (1 - 1/e + o(1))N Unit Fractions Unit Fractions unit_fractions/conlon_2024_question_erdos_graham_egyptian_fractions Claim 1: counting inverse pairs Claim 2: cancelling denominator prime powers Conditional entropy below the mean Counting distinct unit fractions Finite entropy identities used in the proof The continuous entropy exponent External inputs to the Egyptian-fraction proof A finite-window alternative for the entropy count The proper symmetric progression reduction Lemma 1: counting below a reciprocal-sum threshold Lemma 2: optimizer and uniform multiplier estimates Lemma 3: Berry–Esseen input Lemma 4: the modular approximation and its needed restriction Lemma 5: the supply of powersmooth denominators Moment bounds for the entropy product law The modular reservoirs and their size Completing the small-denominator remainder Source versions and proof qualifications Theorem 1: the exponential rate for Egyptian-fraction counts Theorem 2: short modular sums of reciprocals Theorem 3: the precise CFP interface Theorem 4: uniform absorption to an exact rational sum unit_fractions/liu_2024_further_questions_regarding_unit_fractions An admissible carrier for the counting Fourier argument The period obstruction in the printed Proposition 3.2 Lemma 3.3: a dense admissible denominator set Proposition 3.2: a sufficient actual-period form Theorem 1.2: asymptotic count of unit subsums unit_fractions/steinerberger_2024_problem_involving_unit_fractions Lemma — the split-product estimate Counting subsets and signed harmonic sums Signed reformulation and exponential moment Theorem — an eventual bound for unit-fraction subsets The elementary lower bound from the upper half

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