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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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    $\begingroup$ 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. $\endgroup$ – Hagen von Eitzen Dec 27 '12 at 21:43
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    $\begingroup$ "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. $\endgroup$ – Thomas Andrews Dec 27 '12 at 21:47
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    $\begingroup$ 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. $\endgroup$ – Arthur Dec 27 '12 at 21:48
  • $\begingroup$ But this is a possible consequence of the proof. What, if any, are the consequences of the theorem itself? $\endgroup$ – marty cohen Dec 28 '12 at 0:56
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    $\begingroup$ 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)? $\endgroup$ – KCd Dec 28 '12 at 3:43
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A proof of the Riemann Hypothesis will have security implications if that proof can be used to determine the distribution of primes. It is often claimed that the proof for the RH will verify the Goldbach Conjecture but this has not been proven.

I assume that this is where the idea that GC will have security implications comes from, since it is often presented with a confused relationship with the RH, but this doesn't necessarily mean that a proof of the GC will have any real consequences for mathematics.

Much like the GC, proving the RH will not have any major impact on mathematics aside from the phrase "assuming the Riemann Hypothesis" being removed from academic literature.

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