Sixth Prime Came on Schedule

[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.

For instance, I predict that in order to Sieve with the number 5359 down to where 26449 Proth tests sufficed to find 5359•2^5054502+1 is prime (as was done), it was necessary to sieve out all potential factors below around 130 billion, where the sieve eliminates all exponents 'n' such that 5359*2^n+1 is divisible by a number below 130 billion. How close is this to reality?

I wrote my own little sieve. On a 3GHz machine, with any one of the 17 numbers, it takes about a day to sieve out all exponents (up to 20 million) for which there is a factor below 1 billion. I'm sure your sieve is probably faster, but if there is anywhere to find the theory then I can see if any of my math tricks were new (and potentially useful) or not.