Asymptotics for Magic Squares of Primes

TL;DR

This paper establishes asymptotic formulas for counting prime-entry magic squares of size n, extending Green, Tao, and Ziegler's work, and analyzes the complexity of the associated systems of linear forms.

Contribution

It provides the first asymptotic counts for prime magic squares of size n and determines their complexity in the Green-Tao sense, including explicit results for n=3 and n=4.

Findings

Asymptotic formulas for prime magic squares as N→∞

Complexity analysis of the defining systems of linear forms

Equivalence of asymptotics for magic squares with distinct entries

Abstract

Based on the work of Green, Tao and Ziegler, we give asymptotics when $N \to \infty$ for the number of $n \times n$ magic squares with their entries being prime numbers in $[0,N]$. For every $n \ge 3$ we give appropriate systems of linear forms (or equivalently basis) describing all $n \times n$ magic squares with integer entries and we calculate the complexity of these systems in the Green and Tao sense. We compute the precise asymptotics for the cases $n=3$ (complexity 3) and $n=4$ (complexity 1), and the given algorithm works for $n \ge 5$ (complexity 1). Finally, we show that the asymptotics are exactly the same if we impose that all the entries of the magic squares have to be different.