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.).
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.