On the existence of magic squares of powers
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.09364
- arXiv:
- arXiv:2406.09364
- Bibcode:
- 2024arXiv240609364R
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics;
- 11D45;
- 05B15;
- 11D72;
- 11G35;
- 11P55
- E-Print:
- Updated version features revised algorithm making computer search unnecessary as well as proper credit for the work of Flores