# Why is Goldbach's conjecture not included in the millenium prize problems

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?

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This is a question about the person that set the Millenium prize problems, not the Goldbach conjecture. – user50336 Nov 30 '12 at 4:41
In case anyone is not familiar with these problems, the homepage is here: claymath.org/millennium – Gerry Myerson Nov 30 '12 at 5:07
Ask the Clay Institute. When you post a prize, you can post it for any problem you want. I suspect the setters got a lot of advice of what prizes to post, but fundamentally it is their decision. – Ross Millikan Nov 30 '12 at 5:11

The other problems are more important. They have wide-ranging consequences, which Goldbach doesn't.

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Though you are clearly a better mathematician than I, and so your opinion carries more weight (including with me!) than my own, I'm not sure we can take your statement as necessarily true a priori. Imagine, for example, that a proof of the Goldbach conjecture exposes some truth about primes which ends up proving also the Reimann Hypothesis — clearly, in that [extreme] hypothetical situation, one could easily defend the position that Golbach does have "wide-ranging consequences". – Kieren MacMillan Oct 2 '14 at 13:40
@Kieren, Goldbach (or, rather, its proof) could have wide-ranging consequences, but the others do have wide-ranging consequences. – Gerry Myerson Oct 3 '14 at 0:05

Good observation! Just a few remarks:

1. 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.

2. 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.

3. 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.

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Fermat's last theorem was proved before the list was compiled. – Artem Nov 30 '12 at 4:51
What about sieve methods? – Lior B-S Nov 30 '12 at 7:21
"Pure" sieve methods produce only upper bounds in Goldbach as far as I am aware. Although technically you could replace the circle method by more sieve-theoretic ideas. – blabler Nov 30 '12 at 7:24