An interesting variation: if you have the first n primes, partition them in two groups and take the product of all primes in each group. Then take the difference of the two products. This number will not be divisible by any of the primes, and if it is less than the square of the nth prime (and greater than 1) it will be prime. Searching for the partition that produces the lowest difference of products produces the next prime for a short while:

2x5 - 3 = 7

3x7 - 2x5 = 11

11x5 - 2x3x7 = 13

This fails to produce the next prime very soon, but it's slightly interesting.