On the existence of magic squares of powers
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910588906504192 |
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| author | Rome, Nick Yamagishi, Shuntaro |
| author_facet | Rome, Nick Yamagishi, Shuntaro |
| contents | For any $d \geq 2$, we prove that there exists an integer $n_0(d)$ such that there exists an $n \times n$ magic square of $d^\text{th}$ powers for all $n \geq n_0(d)$. In particular, we establish the existence of an $n \times n$ magic square of squares for all $n \geq 4$, which settles a conjecture of Várilly-Alvarado.
All previous approaches had been based on constructive methods and the existence of $n \times n$ magic squares of $d^\text{th}$ powers had only been known for sparse values of $n$. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of magic squares of powers Rome, Nick Yamagishi, Shuntaro Number Theory Combinatorics 11D45, 05B15, 11D72, 11G35, 11P55 For any $d \geq 2$, we prove that there exists an integer $n_0(d)$ such that there exists an $n \times n$ magic square of $d^\text{th}$ powers for all $n \geq n_0(d)$. In particular, we establish the existence of an $n \times n$ magic square of squares for all $n \geq 4$, which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of $n \times n$ magic squares of $d^\text{th}$ powers had only been known for sparse values of $n$. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity. |
| title | On the existence of magic squares of powers |
| topic | Number Theory Combinatorics 11D45, 05B15, 11D72, 11G35, 11P55 |
| url | https://arxiv.org/abs/2406.09364 |