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

TL;DR

The paper proves that the smallest denominator not appearing in any $k$-term distinct Egyptian fraction decomposition of $1$, denoted $v(k)$, satisfies $v(k)\ge e^{c k^2}$ for some absolute constant $c>0$. The proof combines monotonicity properties of $F(k)$ and $D_k$, an injection from $S_k$ to $S_{k+1}$, and a Vose-type decomposition lemma that expresses fractions $a/b$ as sums of unit fractions with a controlled number of terms. By applying this framework to $(m-1)/m$ and analyzing whether $m$ occurs among auxiliary terms, the authors show that any $m<e^{c k^2}$ must appear as a denominator in some $k'\le k$ term decomposition, forcing the lower bound. The work also discusses connections to the $N(b)$ problem and notes potential implications: improvements on $N(b)$ could imply even faster (doubly exponential) growth for $v(k)$ if conjectured bounds hold, with a reciprocal link between lower bounds on $v(k)$ and upper bounds on $N(b-1,b)$.

Abstract

Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$ 1=\frac1{n_1}+\cdots+\frac1{n_k}, $$ 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 $k$.

Theorems & Definitions (5)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2: VoseBLMS
• proof : Proof of Theorem \ref{['thm:main_vkeck2']}