Two problems with prime numbers 
Problem 1. Prove that there exists $n\in\mathbb{N}$ such that in interval $(n^2, \ (n+1)^2)$ there are at least $1000$ prime numbers.
Problem 2. Let $s_n=p_1+p_2+...+p_n$ where $p_i$ is the $i$-th prime number. Prove that for every $n$, there exists $k\in\mathbb{N}$ such that $s_n<k^2<s_{n+1}$.

I've found these two a while ago and they interested me. But don't have any ideas.

For the first one, you can prove that there is a positive integer $n$ such that $\pi((n+1)^2 - 1) - \pi(n^{2}) \geqslant 1000$, where $\pi$ is the prime counting function, using the Prime Number Theorem.
For the second one, I believe Bertrand's Postulate may be useful.
