<p>Alright, so maybe I shouldn't have shrunk this question sooo much... I have seen the post on <a href="https://stackoverflow.com/questions/622/most-efficient-code-for-the-first-10000-prime-numbers">the most efficient way to find the first 10000 primes</a>.  I'm looking for <strong>all possible ways</strong>.  The goal is to have a one stop shop for primality tests.  Any and all tests people know for finding prime numbers are welcome.</p>

<p>And so:</p>

<ul>
<li><strong>What are all the different ways of finding primes?</strong></li>
</ul>

The Sieve of Eratosthenes
is a decent algorithm:
Take the list of positive integers 2 to any given Ceiling.
Take the next item in the list (2 in the first iteration) and remove all multiples of it (beyond the first) from the list.
Repeat step two until you reach the given Ceiling.
Your list is now composed purely of primes.
There is a functional limit to this algorithm in that it exchanges speed for memory.  When generating very large lists of primes the memory capacity needed skyrockets.