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.

Solution to the first:
By inspection of the primes, pick $n = 8715$. Note that $n^2 = 75951225 < 75951233$, and $(n+1)^2 = 75968656 > 75968723$.
Now $75951233$ and $75968723$ are primes, with $\ge 1000$ primes between them so we're done. [1]

[1] The $4446857$th prime is $75951233$ and the $4447859$th prime is $75968723$ (source: http://primes.utm.edu/nthprime/index.php). Further, $4447859 - 4446857 = 1002$.
Exercise: Show that $n = 8715$ is the minimum $n$ satisfying the claim.
