It would appear that the mystery of prime gap is of interest to me for the distribution of primes.
It seems that there's a pattern of gaps building up. While there have been many conjectures out there pertaining to the gaps.
If we start with the gaps of odd...2,2,2,2,2,...
For sieving out multiples of 3, we "squeeze" every other 2-consequtive pairs to make it 2,4,2,4,2,4,...Sieving multiples of 5, we "squeeze" another specific yet consistent pattern of consecutive 2-4 to make it 6, such that 2,4,2,4,6,2,6,4,2,4,2,4,6,2,6,4,...
I need to examine such patterns as we sieve higher multiples.
As for the largest prime known to date: 2^57,885,161-1, which has 17,425,170 digits, the number alone requires about 7MB to store, bit-wise. Therefore, it is unlikely for me to implement a way to store all natural numbers to the largest prime, an easy way to perform the sieve. A cleverer way is required.
I don't care much about Mersenne Primes. I am more interested in having a means to find the largest prime, p, such that I should also known all primes < p.
For finding primes: Lucas-Lehmer Test is generally used.
For proving: Perhaps AKS test?