An Equivalence Between Erdős's Square Packing Conjecture and the Convergence of an Infinite Series

2025/06/29 by Singh, Anshul Raj
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2506.23284

openalex publication_date 2025/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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(n²+1) = n for all positive integers n if and only if the sum ∑_{k≥ 1}(f(k²+1)-k) converges. We also show that if f(k²+1) = k, for infinitely many positive integers then f(k²+1) = k for all positive integers.

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