An Equivalence Between Erdős's Square Packing Conjecture and the Convergence of an Infinite Series
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866908724015136768 |
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| author | Singh, Anshul Raj |
| author_facet | Singh, Anshul Raj |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Equivalence Between Erdős's Square Packing Conjecture and the Convergence of an Infinite Series Singh, Anshul Raj Combinatorics 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. |
| title | An Equivalence Between Erdős's Square Packing Conjecture and the Convergence of an Infinite Series |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.23284 |