The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length

Abstract

Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$1=\frac{1}{n_1}+\cdots +\frac{1}{n_k},$$ where $1\le n_1<\cdots <n_k$ are pairwise distinct integers. In their 1980 monograph, Erdős and Graham asked for quantitative estimates on the growth of $v(k)$ and suggested the lower bound $v(k)\gg k!$. In this paper we give the first known improvement and show that there exists an absolute constant $c>0$ such that the inequality $$v(k)\ge e^{c{k}^2}$$ holds for all positive integers $k$.

7 pages. Accepted for publication in Mathematical Proceedings of the Cambridge Philosophical Society; minor revisions following the referee's suggestion