First, let's factorize the given expression as $$3^{2n+1} -2^{2n+1}- 6^n = (3^n-2^n)(3^{n+1}+2^{n+1})$$ When $n>1$, $we have 3^n-2^n>0$ which implies the given number is a composite number. On the other hand, when $n=1$, the given number is $13$ which is a prime number. Therefore the only qualified solution is $\boxed{1}$.