The numerical results are astonding, for example there are 219,400 such representations for 100,000,000. And in fact it seems as the number of possible representations keeps increasing. And there is nobody who seriously believes that this conjecture is false.
It is know that each even integer is a sum of six or fewer primes.
The first real progess was made by Hardy and Littlewood in 1922, they have shown that for a sufficiently large integer the Conjecture of Goldbach is true! But the sad news is that Hardy and Littlewood used the Generalized Riemann Hypothesis. In 1937 the Russian mathematician Vinogradov was to remove the necessatity of using the generalized Riemann Hypothesis and established that all sufficiently large integers are some of three odd primes. But he was unable to decide how large this number is. In 1956 Borozdkin proved that is big enough. And in 1989 this number was reduced to . Hence if it can be checked by hand that all of those numbers are true, then the Goldbach Conjecuture is settled. But it is far to difficult for a man to do. And even if we accept computer results (because I do not accept anything that a computer does and many mathematicians agree) it is still computationally difficult.
Note: I have taken the above facts from my Number Theory book. I have changed the wording to avoid copyright material.