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

TL;DR

This paper investigates the minimal integer not appearing as a denominator in fixed-length Egyptian fraction decompositions of 1, proving exponential growth bounds for this minimal integer.

Contribution

It establishes the first nontrivial lower bound on the growth of the minimal excluded denominator, showing it grows at least exponentially with respect to the square of the number of terms.

Findings

Proves v(k) ≥ e^{c k^2} for some constant c > 0

Improves upon the previous lower bound of v(k) ≫ k!

Provides a new quantitative estimate on denominators in Egyptian fractions.

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, Erd\H{o}s 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 $k$.