Goldbach Conjecture Consequences 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: One result that is needed for Goldbach to be true is that natural number n not divisible by 3, has a prime congruent to it mod 3, between itself and twice itself.
Why ?
...0,1,2,0,1,2,0,1,2,0,1,2, ...
Now if we pick a 1 mod 3 for example we note that any distance 1 mod 3 below it is divisible by 3 and therefore except $3$ itself can't be prime. Likewise,any distance 2 mod 3 has the upper value divisible by 3.  So if we want 2 equidistant primes, either 2n-3 is prime or there is a prime congruent to 1 mod 3 between  the extremes of n and 2n...
Same with 2 mod 3, but with the realization the direction changes as 2 is -1 mod 3 ...
But we can also use modular prime counting function, and pigeonhole principle ...
Why ?
Because for those two remainders modulo 3, now the difference in modular prime counting functions  for the relevant remainder can't exceed the modular composite counting function up to n , if it did we can pigeonhole Goldbach ...
You can state Goldbach with nearly any function outputs, and so have implications on almost any function ...
