An Equivalence Between Erd\H{o}s's Square Packing Conjecture and the Convergence of an Infinite Series

TL;DR

This paper establishes a deep connection between Erdős's square packing conjecture and the convergence of a specific infinite series, providing new insights into the structure of optimal square packings.

Contribution

It proves an equivalence between the conjecture and series convergence, and shows that infinite instances imply the conjecture holds universally.

Findings

Proves the equivalence between the conjecture and series convergence.

Shows that infinitely many solutions imply the conjecture holds for all integers.

Provides a new approach to understanding square packing problems.

Abstract

Let $f(n)$ denote the maximum sum of the side lengths of $n$ non-overlapping squares packed inside a unit square. We prove that $f(n^2+1) = n$ for all positive integers $n$ if and only if the sum $\sum_{k\geq 1}(f(k^2+1)-k)$ converges. We also show that if $f(k^2+1) = k$, for infinitely many positive integers then $f(k^2+1) = k$ for all positive integers.