On a conjecture of Erdős and Graham about the Sylvester's sequence

Abstract

Let $\{u_n\}_{n=1}^{\infty}$ be the Sylvester&#39;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 &#34;all rationals have eventually greedy best Egyptian underapproximations&#34; holds, we establish a generalization of this conjecture using a non-constructive approach. [This paper solves Problem 315 on Bloom&#39;s website &#34;Erdős problems&#34;.]