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## Algebraic aspects of the Goldbach conjecture

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?)

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A minute of googling shows that the weak Goldbach conjecture (or 3-primes problem) has been stated and solved for function fields (Effinger and Hayes "A complete solution to the polynomial 3-primes problem"). The proof uses an adaption of the circle method for function fields as well as a computer check, so definitely fails the "steers away from hard analysis" test. – BR Oct 24 2010 at 22:18

You might be interested in this article on Goldbach over function fields. The approach is rather geometric/algebraic, so it does pass your "steers away from hard analysis" test.

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Here's a paper that might be considered a step in that direction. The authors construct an explicit family of polynomials $(F_N)_{N \in \mathbb{N}}$ such that the $N$th cyclotomic polynomial divides $F_N$ iff $N$ is not the sum of two primes.