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

Title: 论幂幻方的存在性

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: 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. 所有以前的方法都是基于构造性方法，而仅知道$n \times n$阶魔方阵的$d^\text{th}$次幂在$n$的稀疏值下存在。 我们通过 Hardy-Littlewood 圆法证明了我们的结果，在这种情况下，该方法基本上将问题简化为找到系数矩阵列的足够多的不相交的线性无关子集，这些子集定义了魔方阵的方程。 我们为此数量证明了一个最优（常数范围内）的下界。