I'm asking the question on a bit of whim, but I do wonder what answers I would get. The Goldbach conjecture is usually discussed in the realm of the distribution of primes and/or probability. Methods I've seen in the past are mostly analytic.
Have there been methods of attack on this problems that are at their core not about the distribution of primes, and whose methodology steers away from hard analysis? Have there been methods of attack that are completely in the realm of algebra? How have they fared? (for example: was a version of it stated and proven over function fields?)