Asymptotics for Magic Squares of Primes

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.

34 pages, 6 figures

References in corpus (1)

• Six Little Squares and How Their Numbers Grow