  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.
