Ok, here's my attempt to answer many of the questions in this thread. Let's take our latest 13-chain from yesterday as an example:
"time" : "2014-01-20 11:57:59 UTC",
"epoch" : 1390219079,
"height" : 368051,
"ismine" : false,
"mineraddress" : "AeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta",
"primedigit" : 107,
"primechain" : "1CC0d.c48332",
"primeorigin" : "12512390300891276190682243916246636610000954402441740274147501230375694290702478259358177371388272647651840",
"primorialform" : "106680560818292299253267832484567360951928953599522278361651385665522443588804123392*61#"
This is a Cunningham chain of the first kind (1CC). Other possible types are the second kind (2CC) and BiTwin (TWN).
The numbers of a 1CC chain look like this:
origin * 2^i - 1
Here 'i' starts from zero. So the first prime in the chain is origin - 1, the second prime is origin * 2 - 1, and then origin * 4 - 1, etc.
In Primecoin mining the origin is composed like this:
origin = headerHash * primorial * candidateMultiplier
The header hash is an intermediate hash calculated from the block header (it is NOT the final block hash). It's a 256-bit number that is required to be greater than 2^255.
The primorial p# is defined as the product of all primes up until 'p', which means that 61# = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47 * 53 * 59 * 61 = 117288381359406970983270.
The candidate multiplier is a number produced by sieve. It's typically less than 1 million. The sieve uses an efficient algorithm to filter out a large number of prime factors. The sieve rejects all candidates x for which headerHash * primorial * x * 2^i - 1 is divisible by one of the first 8000 prime numbers.
Side note: The "extended" sieve algorithm considers multipliers of the form 2^j * x. So we are multiplying all the candidate numbers by powers of 2. It turns out that these new candidates can be checked efficiently because we are shifting the origin so that the second prime becomes the first prime and the third prime becomes the second prime etc.
I think many people are still wondering what "1CC0d.c48332" means. I already explained that 1CC is a Cunningham chain of the first kind. '0d' is the length of the chain in hexadecimal (that is 13). 'c48332' is the 24-bit representation of the fractional length. We can convert it into the usual decimal notation:
0xc48332 / 0xffffff = 0.7676
So our example chain would meet a difficulty of 13.7676.
Ghashes are not applicable to Primecoin. I'm guessing some websites are trying to interpret the Primecoin difficulty like it would be Bitcoin/Litecoin difficulty, which doesn't produce any sensible results. My Primecoin charts show the average prime chain rate for the network:
Right now it's about 1.94 chains/min. So the total 'chainsperday' of the whole network would be about 2793 chains/day.