On the existence of magic squares of powers

TL;DR

This paper proves that for any fixed power $d \\geq 2$, sufficiently large magic squares of $d^{th}$ powers exist, including all squares of size at least 4 for squares, using advanced number theory methods.

Contribution

It establishes the existence of large magic squares of $d^{th}$ powers for all sufficiently large sizes, solving a longstanding conjecture for squares.

Findings

Existence of $n imes n$ magic squares of $d^{th}$ powers for all large $n$

Proof uses Hardy-Littlewood circle method and linear independence of column subsets

Settles the conjecture for squares of size at least 4

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.