Is there a simple algorithm that can determine if X is prime?

I see that Fermat's primality test has already been suggested, but I've been working through Structure and Interpretation of Computer Programs, and they also give the Miller-Rabin test (see Section 1.2.6, problem 1.28) as another alternative. I've been using it with success for the Euler problems.

Here's a simple optimization of your method that isn't quite the Sieve of Eratosthenes but is very easy to implement: first try dividing X by 2 and 3, then loop over j=1..sqrt(X)/6, trying to divide by 6*j-1 and 6*j+1. This automatically skips over all numbers divisible by 2 or 3, gaining you a pretty nice constant factor acceleration.

The first algorithm is quite good and used a lot on Project Euler. If you know the maximum number that you want you can also research Eratosthenes's sieve.

If you maintain the list of primes you can also refine the first algo to divide only with primes until the square root of the number.

With these two algoritms (dividing and the sieve) you should be able to solve the problems.

Edit: fixed name as noted in comments

To generate all prime numbers less than a limit Sieve of Eratosthenes (the page contains variants in 20 programming languages) is the oldest and the simplest solution.

In Python:

def iprimes_upto(limit):
    is_prime = [True] * limit
    for n in range(2, limit):
        if is_prime[n]:
           yield n
           for i in range(n*n, limit, n): # start at ``n`` squared
               is_prime[i] = False

Example:

>>> list(iprimes_upto(15))
[2, 3, 5, 7, 11, 13]