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

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 $$\underset{n\to \infty }{\mathrm{liminf}}{a}_n^{\frac{1}{2^n}}<\underset{n\to \infty }{\lim }{u}_n^{\frac{1}{2^n}}=c_0=1.264085\dots .$$ 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".]

23 pages; v2 generalizes the previous results; v3 fixes some typographical errors and adds several remarks; v4 corrects the definition of underapproximation and adds a final section proposing several open problems

References in corpus (2)

• Underapproximation by Egyptian fractions
• On eventually greedy best underapproximations by Egyptian fractions