On a square packing conjecture of Erdős

TL;DR

Let $f(n)$ be the maximum total side length of $n$ non-overlapping squares packed inside a unit square; the paper shows that Erdős's conjecture $f(n^2+1)=n$ is equivalent to the convergence of the series $\sum_{k\ge1}(f(k^2+1)-k)$. It achieves this via a unifying inequality $a f(m) \le a^2 - b^2 + b f(b^2 - a^2 + m)$ and a downward-propagation analysis of $\epsilon(k)=f(k^2+1)-k$. The same argument extends to the unit equilateral triangle and to parallelograms by tiling-based constructions, implying the conjecture or its failure would be reflected in the same series convergence criterion. These results unify the square, triangle, and parallelogram packing problems under a single analytic framework and offer a clear criterion for proving or refuting Erdős's conjecture.

Abstract

Let $f(n)$ be the maximum sum of the sides of non-overlapping squares (or equilateral triangles) packed inside a unit square or (unit equilateral triangle). In this paper, we explore some properties of $f$ and examine how the square and triangle cases are similar. We prove that a conjecture of Erdős, which says that $f(k^2+1) = k$ for all $k$, is equivalent to the convergence of the series $\sum_{k\geqslant 1}(f(k^2+1)-k)$. We also explore the case of parallelograms and discuss how that is similar to the case of unit square and triangle.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof : Proof of Theorem 2