Title: On the existence of magic squares of powers Show Chinese title

Title: 关于幂幻方的存在性

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 V\'{a}rilly-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. Show Chinese abstract

Abstract: 对于任意的$d \geq 2$，我们证明存在一个整数$n_0(d)$，使得对所有$n \geq n_0(d)$，都存在一个$d^\text{th}$次幂的$n \times n$阶幻方。 特别是，我们建立了对所有$n \geq 4$存在一个$n \times n$阶平方幻方的存在性，这解决了 Várilly-Alvarado 的一个猜想。 所有先前的方法都基于构造性方法，且仅对于稀疏的 $n$ 值，$n \times n$ 阶 $d^\text{th}$ 次幻方的存在性才被知晓。 我们通过哈代-利特尔伍德圆法证明了这一结果，在这种设定下，该问题本质上归结为找到足够数量的系数矩阵列的互不相交的线性无关子集。 我们为此量证明了一个最优（至多常数因子）的下界。