jmc

Let $n$ be any integer that has its square between 4000 and 7000. Then $63<n<84$, because $63^2<4000<64^2$ and $83^2<7000<84^2$. Between 63 and 84, the only primes are 67, 71, 73, 79, and 83. Thus, the answer is $\boxed{5}$.