As we all know, the Goldbach's Conjecture is one one of the oldest and best-known unsolved problems in mathematics.
I was going through some of the attempts made to solve it and got fascinated as to why it hasnt been included in the Millenium prize problems?
On
Good observation! Just a few remarks:
First of all, as previous poster said, there is a lack of tangible consequences. Note that Fermat's last theorem was also not on the list.
Secondly, we can essentially solve the ternary goldbach problem. The ternary Goldbach problem says that every odd integer is the sum of at most three primes. This was shown to be true for all sufficiently large integers by Vinogradov in the late 30's. A recent pre-print of Helfgott (http://arxiv.org/abs/1205.5252) makes people hopeful that very soon this will be settled for all the integers.
Finally, the array of techniques for attacking Goldbach's problem is not very vast. The circle method was and still is the best candidate. We understand its limitations fairly well, and know exaclty why we can't prove the full Goldbach problem. Note however that it is known that almost all even integers are sums of two primes.
The other problems are more important. They have wide-ranging consequences, which Goldbach doesn't.