Problem 315

Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$ , so that $\sum_{k\geq 1}\frac{1}{u_k+1}$ and $u_k=\lfloor c_0^{2^k}+1\rfloor$ for $k\geq 1$ , where

c_0=\lim u_n^{1/2^n}=1.264085\cdots.

Let $a_1<a_2<\cdots$ be any other sequence with $\sum \frac{1}{a_k}=1$ . Is it true that

\liminf a_n^{1/2^n}<c_0=1.264085\cdots?

Source: erdosproblems.com/315

Number theory
Unit fractions

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved, by two independent sources. Li and Tang's Corollary 1.7 (arXiv:2503.12277, March 2025) proves exactly the statement: every strictly increasing sequence of positive integers other than Sylvester's with reciprocal sum $1$ has $\liminf a_n^{1/2^n}<c_0=1.264085\ldots$ . Kamio's Theorem 8 (arXiv:2503.02317, March 2025; an author preprint) proves it for nondecreasing sequences and for every unit fraction $1/n$ in place of $1$ . The two preprints appeared eleven days apart: Li and Tang's Remark 1.13 records Kamio's proof as independent, and Kamio's, the earlier one, does not cite theirs. Neither proof has a refereed publication: Li and Tang's Acta Math. Hungar. paper (177 (2025), 41--63) publishes their conditional generalization and cites the preprint for the proof of this statement. Kovač and Tang's 2026 preprint generalizes the statement to rationals and re-derives it by a non-constructive route; it is a pending claim. The site's label is PROVED (LEAN); its Lean qualifier is a catalog label whose scope is qualified under Formalization and the Lean label below, and no local kernel credit is claimed. The two accepted claims are recorded on Li and Tang's claim page (2025) and Kamio's claim page (2025), and the pending claim on Kovač and Tang's claim page (2026).

LiTa25: Li, Z. and Tang, Q., On a conjecture of Erdős and Graham about the Sylvester's sequence. arXiv:2503.12277 (v1 15 March 2025; v4 21 March 2025, 23 pages). Conjecture 1.3, p. 3; Corollary 1.7, p. 4 of the preprint. The arXiv listing's journal reference points to the authors' paper Generalizing a conjecture of Erdős and Graham via best Egyptian underapproximations, Acta Math. Hungar. 177 (2025), no. 1, 41--63, DOI 10.1007/s10474-025-01566-8, online 13 October 2025 (Crossref record accessed). That paper is not held; its published abstract says that the conjecture was resolved constructively by Kamio and independently by the authors and that the paper proves a generalization assuming the eventually-greedy claim (Theorem 1.6 there, Theorem 1.9 of arXiv v4), and its reference list cites arXiv:2503.12277 as a separate item, so it is not a refereed publication of Corollary 1.7. Library home: Li and Tang 2025; result page Li and Tang 2025, Corollary 1.7.
Ka25: Kamio, Y., Asymptotic analysis of infinite decompositions of a unit fraction into unit fractions. arXiv:2503.02317v1 (4 March 2025, 5 pages, the only version). Preprint. Theorem 8, p. 3. Library home: Kamio 2025; result page Kamio 2025, Theorem 8.
KoTa26: Kovač, V. and Tang, Q., Eventually greedy best Egyptian underapproximations of rational numbers via optimal control. arXiv:2607.28387 (v1 30 July 2026; v2 4 August 2026, 30 pages). Preprint; Corollary 3 (p. 4) and Theorem 4 (p. 5) generalize this problem to rationals; claim page Kovač and Tang's claim page (2026). Library home: Kovač and Tang 2026.
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. 41. Library home: Erdős and Graham 1980.
OEIS: Sequence A000058, Sylvester's sequence $2,3,7,43,1807,\ldots$ (N. J. A. Sloane), and Sequence A076393, the decimal expansion of the Vardi constant $1.2640847353\ldots$ (B. Cloitre, 2002), with the comment that Vardi showed $\text{A000058}(n)=\lfloor c^{2^{n+1}}+1/2\rfloor$ ; both accessed.

Problem 315

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