Let $n$ be a positive integer , $A_n$ the $n\times n$-matrix filled with the first $n^2$ primes from left to right and from top to bottom, for example $$A_4=\pmatrix{2&3&5&7\\11&13&17&19\\23&29&31&37\\41&43&47&53}$$ and $D_n=|\det(A_n)|$
Can $\omega(D_n)$ , where $\omega(n)$ denotes the number of distinct prime factors of $n$, be arbitary large ? In other words, can the determinant of the matrix have arbitary many distinct prime factors ?
I found out $\ \color \red{\omega(D_{36})=11}\ $ and $\ \color\red {\omega(D_{122})>11}\ $. The determinants soon get very large in absolute value , potentially allowing many prime factors, but I am not sure whether arbitary many are possible.