On a conjecture of Erdős and Graham about the Sylvester's sequence
Zheng Li, Quanyu Tang
TL;DR
The paper addresses Erdős and Graham’s conjecture comparing the growth rate of any unit-reciprocal sequence {a_i} with that of Sylvester’s sequence {u_n}, showing liminf_{n→∞} a_n^{1/2^n} is strictly smaller than lim_{n→∞} u_n^{1/2^n}=c_0≈1.264085 when ∑ 1/a_i=1. It provides a constructive proof by explicitly building an eventually Sylvester sequence {c_n} that satisfies the same unit-sum constraint and lies below the Sylvester sequence in the 2^−n-exponent metric. It also develops a conditional, non-constructive generalization relying on a conjectured property of greedy best Egyptian underapproximations (Conjecture 2), leading to existence results for a distinct {c_n} and analogous inequalities. Together, these results advance our understanding of unit-fraction representations, Egyptian greedy algorithms, and the Erdős–Graham problem landscape, with connections to recent independent work and open questions about characterizing when greedy best underapproximations are unique.
Abstract
Let $\{u_n\}_{n=1}^{\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $ a_1 < a_2 < \cdots $ be any other positive integer sequence satisfying $ \sum_{i=1}^\infty \frac{1}{a_i} = 1 $. In this paper, we solve a conjecture of Erdős and Graham, which asks whether $$ \liminf_{n\to\infty} a_n^{\frac{1}{2^n}} < \lim_{n\to\infty} u_n^{\frac{1}{2^n}} = c_0 = 1.264085\ldots. $$ We prove this conjecture using a constructive approach. Furthermore, assuming that the unproven claim of Erdős and Graham that "all rationals have eventually greedy best Egyptian underapproximations" holds, we establish a generalization of this conjecture using a non-constructive approach. [This paper solves Problem 315 on Bloom's website "Erdős problems".]