In article <1140731666.341436.280...@z34g2000cwc.googlegroups.com>,
Find all the odd primes between 0 and 200,000,000.
> I was wondering what are the current methods for verifying Goldbach's
> Conjecture up to a certain number?
To verify Goldbach's conjecture up to 200,000,000:
Make a sieve of all even numbers from 4 to 200,000,000.
Remove 4 because 4 = 2 + 2.
Remove 3+p for all primes p <= 200,000,000 - 3 from the sieve.
Remove 5+p for all primes p <= 200,000,000 - 5 from the sieve.
Remove 7+p for all primes p <= 200,000,000 - 7 from the sieve.
Remove 11+p for all primes p <= 200,000,000 - 11 from the sieve.
and so on, until the sieve is empty. If the sieve gets empty, GC is
verified up to 200,000,000, otherwise it is false.
Now repeat the following with M = 200,000,000, M = 300,000,000, M =
400,000,000 etc. etc:
Create a sieve with all even numbers from M+2 to M+100,000,000.
Find the largest prime p < M. Remove q + p from the sieve for all primes
q, M-p < q <= M-p + 100,000,000, taking the primes q from the sieve.
Replace p with the next smaller prime, and again remove primes of the
form q+p from the sieve.
Repeat until either the sieve is empty (GC has been verified for values
M < N <= M + 100,000,000) or until the nextsmaller prime p is less than
M - 100,000,000 (verification has failed, needs a closer look).
That should be reasonably fast. You can use a sieve to find all primes p
such that M - 100,000 <= p < M; that is likely enough.