Li 2025 conjecture erdos graham about sylvester s

../

corollary_1_7: States that every increasing sequence of positive integers other than Sylvester's with reciprocal sum 1 has liminf of a_n^(1/2^n) strictly below the Vardi constant 1.264085..., the question of Problem 315.

theorem_1_5: States that an eventually Sylvester sequence of positive reals with reciprocal sum 1, following the recurrence from an index N at least 2 on, whose reciprocal sum over the first N-1 terms is below Sylvester's, has liminf of a_n^(1/2^n) equal to its limit and strictly below Sylvester's.

theorem_1_6: States that for every increasing integer sequence other than Sylvester's with reciprocal sum 1 there is an eventually Sylvester sequence of positive reals meeting the hypotheses of Theorem 1.5 whose limit of c_n^(1/2^n) is at least the liminf of a_n^(1/2^n).

theorem_1_9: States that, assuming the Erdős–Graham claim that every rational in (0,1] has eventually greedy best Egyptian underapproximations, for a rational lambda with unique best m-term underapproximations every other increasing integer sequence with reciprocal sum lambda has a smaller liminf of a_n^(1/2^n).

---

Zheng Li, Quanyu Tang, On a conjecture of Erdős and Graham about the Sylvester's sequence. arXiv:2503.12277 (2025).

The copy read for this card is arXiv:2503.12277v4 (21 March 2025), 23 pages (v1 15 March 2025, v2 18 March 2025, v3 20 March 2025 per the listing's submission history; the arXiv comment says v4 corrects the definition of underapproximation and adds a final section of open problems). The arXiv journal reference and DOI point to Z. Li and Q. Tang, 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, published online 13 October 2025 (the Crossref record). That paper is not this preprint under a new title: its abstract says the conjecture was resolved constructively by Kamio and independently by the authors and that the paper treats the non-constructive generalization, and its reference list cites arXiv:2503.12277 as a separate item (the Semantic Scholar and Crossref records). The published text was not obtained or compared, and the labels below are the preprint's. Read status: claims checked. Conjecture 1.3, Definition 1.4, Theorems 1.5 and 1.6 and Corollary 1.7 (pp. 3--4) were read clause by clause on the rendered page image of p. 4 and the text layer of pp. 3--4 on 2026-09-18, and the corollary's deduction (p. 19) was read. On 2026-10-08 the page images of pp. 1--23 were read: the statements of Section 1 (Conjecture 1.8, Theorems 1.9, 1.11, 1.12, Corollary 1.10) and Section 5 clause by clause, and the proofs of Sections 3 and 4 for structure only; nothing is verified. Result pages: corollary_1_7, theorem_1_5, theorem_1_6 and theorem_1_9. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2503.12277), every other right reserved.

The paper proves a conjecture of Erdős and Graham, its Conjecture 1.3: with Sylvester's sequence $u_1=2$, $u_{n+1}=u_n^2-u_n+1$, every other increasing sequence of positive integers $a_1<a_2<\cdots$ with $\sum1/a_i=1$ has $\liminf a_n^{1/2^n}<\lim u_n^{1/2^n}=c_0=1.264085\ldots$, the Vardi constant (Corollary 1.7). The proof is constructive and has two halves. Theorem 1.5 shows that an eventually Sylvester sequence of positive reals (the recurrence condition of Definition 1.4, which the print states for integers) with reciprocal sum 1, following the recurrence from an index $N\ge2$ on, whose reciprocal sum over the first $N-1$ terms is below Sylvester's, has a strictly smaller limit of $a_n^{1/2^n}$. Theorem 1.6 constructs such a sequence $c_n$ for any integer competitor with $\liminf a_n^{1/2^n}\le\lim c_n^{1/2^n}$; its print fixes $N\ge2$ in advance, while the construction yields one $N$ depending on the competitor, which is all the corollary uses. A second, non-constructive route (Theorem 1.9, through Theorems 1.11 and 1.12) proves a generalization to rationals $\lambda\in(0,1]$ with unique best $m$-term Egyptian underapproximations, conditional on Conjecture 1.8, the paper's restatement of the Erdős--Graham claim that every rational has eventually greedy best Egyptian underapproximations, which Graham later posed as a question. Section 5 states the unconditional version as Conjecture 5.1 and asks for a proof or disproof of Conjecture 1.8 (Problem 5.3). The tools are greedy Egyptian underapproximation, the identity $\sum_{i<n}1/u_i+1/(u_n-1)=1$, and a monotonicity lemma for the limit of an eventually Sylvester sequence in its starting term (Lemma 2.11).

Source: https://arxiv.org/abs/2503.12277.

Bears on.

• #315: Corollary 1.7 proves the paper's Conjecture 1.3, the problem's question with the excluded sequence written as Sylvester's $2,3,7,43,\ldots$; Theorems 1.5 and 1.6 are its two halves, and Theorem 1.9 generalizes it to rationals $\lambda\in(0,1]$ whose best $m$-term Egyptian underapproximation is unique for every $m$, conditionally on Conjecture 1.8.
• #206: Conjecture 1.8 asserts the eventually-greedy property that the problem asks about for almost every $x$, but for every rational in $(0,1]$; it is a rational variant, not the problem. Theorem 1.9 assumes it, and the paper proves nothing toward it.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.