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.).
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.