Is there any hope to disprove Goldbach's conjecture? It is widely believed, that Goldbach's conjecture is true. But suppose, there is a 
counterexample of, lets say, 50 digits. Is there any hope to prove this counterexample
to be one ? Brute force surely would not work. Any ideas ?
 A: The brute force method is effective in this case.  Suppose I claim 37998938 is a counterexample to Goldbach. As it happens, the lowest prime that splits 37998938 is 1039.  This is the smallest Goldbach-unfriendly number for all primes under 1000.  3325581707333960528 needs the prime 9781 for a Goldbach split.
What is the smallest Goldbach-unfriendly number for all primes under a million?  That would be very hard to find, but very easy to check.
For the hypothetical 50-digit counterexample, it would take a few minutes to brute-force verify that the number was an amazing find, working for all primes under a billion.  From there, portions would be farmed out for more brute force checking.  If the counterexample survived the first hours of incomplete checking, it would become a highly studied number, and would not be ignored.
So far, though, using brute force with primes under 10000 has worked in all cases. Finding an example requiring a prime over 10000 would be a publishable accomplishment.
