TITLE: Finding the 10001th Prime
QUESTION [11 upvotes]: I'm helping my son with Project Euler and we're working on problem 7, "What is the 10001st prime number?"  We'll use a Sieve of Eratosthenes and we'll increase the size of the initial array until we're left with 10001 primes.  We'll start with a pretty big array and increase it by whatever seems reasonable, since there is no time constraint, until we get the answer.
My question is, is there a way to make an informed guess about the size of the initial array?

REPLY [14 votes]: Wikipedia gives the bounds $n \ln n + n(\ln\ln n - 1) < p_n <  n \ln n + n \ln \ln n$, where $p_n$ is the $n$th prime. So the $10001$st prime is between 104318 and 114320.