Originally posted by kuroyama1
Can anyone tell me where to find some theory, such as where the 6th prime would be found? I've got a few theory ideas myself, but don't feel like repeating what others have said.
These is some discussion about models for SOB estimates in the thread on A Resource Allocation Model and the thread on A New Estimate for finding Primes above 3M. Be aware that late in the second thread we conclude the first part is mostly rubbish - a poor and awkward estimate of Proth Weights that can more easily be looked up in Gallot's paper or calculated with Brennan's applet (they differ by a factor of 2 because they use different definitions of weight).

The core of these estimates is
1. The probability that any one value of (k,n) is prime is a proth-weight adjusted Prime Number Theorem value
2. For a fixed K, the expected number of primes from n1 to n2 is the sum of the probabilities in part 1, which is adequately approximated by the integral.
3. If the expected number of primes in a range is "x", the Poisson approximation gives a good estimated of the probability there are zero (or one or two or three) primes in the range. For zero, this is exp(-x).
4. The k values are independent, so the probability of "i" primes can be found by combining the the probability of zero or not-zero primes for each k (we don't care about 2 or 3 primes for the same k).

Note that sieving has no impact on the range where primes will be found, only on the amount of time it will take to find them. A Proth-weight adjusted Merten's Theorem works well for estimating the number of factors to be found by sieving.

William