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

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.