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I have been looking into the Goldbach Conjecture pretty recently and I have often heard that it would have far-reaching consequences. However, I haven't found many of the actual consequences. I was wondering if you all could supply me with some of these consequences (theorems, etc.).

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Assume someone proved or disproved the Goldbach conjecture. Main consequence: Everybody who has used the Goldbach conjecture as an example of a presumed undecidable statement would have to edit their work. – Hagen von Eitzen Dec 27 '12 at 21:43
"I have often heard that it would have far-reaching consequences." I have never heard this. Can you cite an example? What I have heard is that any proof would almost certainly require some technical breakthrough that might help with other problems like the ABC conjecture. – Thomas Andrews Dec 27 '12 at 21:47
Just like the abc-conjecture / theorem, Fermat's last theorem and Szemerédi's theorem, a proof of the Goldbach conjecture is likely to contain heaps of new and probably interesting theory. – Arthur Dec 27 '12 at 21:48
But this is a possible consequence of the proof. What, if any, are the consequences of the theorem itself? – marty cohen Dec 28 '12 at 0:56
As with Fermat's Last Theorem, the statement of Goldbach's conjecture on its own (not any potential proof) doesn't have amazing consequences. The problem is famous because it is old and elementary to state. Perhaps you misunderstood what was meant when you heard it would have great consequences, or whoever made this claim was misinformed. Where did you hear such a remark (often)? – KCd Dec 28 '12 at 3:43

Suppose the proof exists. It's either a constructive proof or an existential proof.

Now consider a googolplex (10^10^100). If the Goldbach proof is constructive, then then we can apply the constructive method to split a googolplex into two prime numbers, at least one of them much larger than the current largest-known prime. We could generate large primes of any size.

If it's existential, then the methods could still be applied to a googolplex, and perhaps a range could be given for where (googolplex-prime) is prime. It would still lead to useful tools in prime number research.

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Huh? Consider the theorem that there are infinitely many primes, hence there is a prime greater than googolplex. The proof that there is such a prime doesn't lead to useful ways of generating primes or useful tools in research. – KCd Dec 29 '12 at 1:32
The Goldbach Proof would in some way concretely prove that a googolplex could be split into 2 primes. That would be useful for prime generation. – Ed Pegg Dec 29 '12 at 1:40
Why would a proof of Goldbach's conjecture in some way concretely prove that? It's already known that every odd number above $10^{2000}$, say, is a sum of three odd primes (see Wikipedia page on the weak Goldbach conjecture), but this doesn't lead to useful ways of generating primes. – KCd Dec 29 '12 at 1:44
A constructive Goldbach Proof might split a googolplex into two primes automatically. Thus, the quibble is whether an existential Goldbach proof would be useful. Vinogradov's theorem is existential, but we can use it to make a lower bound on the minimal number of 3-prime splits for googolplex+1. That's not very useful. The usefulness of an existential proof would depend on its nature. – Ed Pegg Dec 29 '12 at 2:02
I don't believe there will be a proof of Goldbach's conjecture that is constructive. Look at progress on the weak Goldbach conjecture: not constructive. Look at Dirichlet's theorem that when $(a,m) = 1$ there are infinitely many primes $p \equiv a \bmod m$: there is no known proof of this theorem (in its full generality, not for specific choices like $a=1$ and $m=4$) that involves writing down a formula for such a prime. – KCd Dec 29 '12 at 3:20

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