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So this was my experiment once when influenced to play with primes a couple hours from a math class.

To the left is a graph of a particular distribution of the first 1900 primes.

PrimeUtil CLASS:

Stores a growing list of known primes.

Can check if a given integer is a prime.

Can retrieve the i'th prime.

Can determine the result of the pi function (prime-counting function) of a given double.

Updates list of primes as needed for queries.

Checking a candidate for the next prime (to be added to growing list) has running time: n (the size of the growing list of known primes)

Determining an integer is prime has running time: lg n (binary search) *

Evaluating pi has running time: lg n (binary search) *

*not including running time of finding next primes when list is insufficient.

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