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. – B R Oct 24 '10 at 22:18
The circle method is about as analytic as one can get over function fields. – Greg Martin Jan 6 '14 at 23:40

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.

http://cgi.cecm.sfu.ca/~pborwein/PAPERS/P235.pdf

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My question Integer Solutions of $x+y^n = y + x^m$ for $n < m$ tries to build a link between Goldbach Conjecture and Algebraic Number Theory.

Two questions can be raised regarding Goldbach's Conjecture: for any given even number $v > 6$, does exist there a procedure in theory to find a pair of primes such that $v$ is the sum of those two primes? and what happens to the even number $v$ if it is not a sum of any two primes? For the first question we will do the following: first, we pick a prime $p < v - 2$ that does not divide $v$. If $v-p$ is a prime, then we can stop here. Otherwise, $v - p$ is a product of at least 2 primes and none of them divides $v$. More specifically, $p \nmid v$ and $$v - p = \prod_{i = 1}^n p_i$$ is a product of some primes $p_1, p_2, \cdots, p_n$, and $p_i \nmid v$ for $i = 1, 2, \cdots, n$. Let $v_i = v - p_i$ for $i = 1, 2, \cdots, n$. Next, if any $v_i$ is prime, we stop here. Otherwise we assume each $v_i$ is a product of at least two primes not dividing the even number $v$: $$v - p_i = \prod_{j = 1}^{n_i} q_{ij}$$ Thus, we can do the same thing for each $q_{ij}$ as we did for $p_i$. Now assume we keep doing this procedure for a long time, then eventually either we find a pair of primes that $v$ is the sum of them or we have a finite set of primes $R$ such that for each $p\in R$, $v-p$ is a product of primes in $R$. Now we are ready to answer the second question. We assume that $v$ is not a sum of any two distinct primes. Then it is clear that the set $R$ is finite since there are only finitely many primes $p$ such that $p < v$ and $p\nmid v$ for any given $v$. Mathematically, we can describe the procedure as follows. Given an even number $v$ and a finite set of primes $R$, we have $p\nmid v$ for any $p\in R$ and $v - p$ is a product of at least 2 primes in $R$. If we specify $R=\{p_1,p_2,\cdots,p_n\}$ and introduce a non-negative integer matrix $A=(a_{ij})$ of order $n$, then we are able to describe the related numbers in the following formula $$v - p_i = \prod_{j=1}^n p_j^{a_{ij}} \mbox{ or } v =p_i + \prod_{j=1}^n p_j^{a_{ij}}$$ with condition $$\sum_{j=1}^n a_{ij} \geq 2$$ for $i = 1, 2, \cdots, n$. Our aim is to name this even number $v$ along with the finite set of primes $R$ with algebraic equations a numerical pipe or pipe for short.

Readers may notice that the definition of a pipe requires an assumption that there exist an even number being not a sum of two distinct primes. In fact it is easy to get rid of this unrealistic assumption if we know what a pipe should look like. We can start with any even number $v > 6$ and a prime $p < v - 2$ not dividing $v$. If $q = v - p$ is a prime, then we are done with this procedure. If not, then $q$ is a product of at least two primes not dividing $v$. Now we pick a new prime $p$ from the set of prime divisors in $q$, and repeat the procedure as we just have done for the old $p$. As we know, this procedure will end up with two cases: we find a pair of primes that $v$ is the sum of them or the second exceptional case: we get a pipe!

We do have pipes. The example is a pipe with $v = 2200$ and $R=\{3, 13\}$:
$$2200 = 3 +13^3 = 13 + 3^7$$

We call it Pipe Euler for its elegance and simplicity. It is a surprise that there are only six pipes of this kind for the even number $v < 500,000,000$ and they are listed in the chapter for basic pipes. Each of them is named in order of its value after a great mathematician: Archimedes, Bernoulli, Cauchy, Dirichlet, Euler and Fermat. If there is one pipe after the six pipes, we reserve the name Gauss for it. We believe that there are only finite number of pipes of this kind.

It seems that we are ready to present a formal definition on pipe. But we can not do any work if we only have six pipes on the pipe world. We just need more. We need a definition of pipe board enough that we have more pipes to work on instead of just six pipes. For this reason, we allow that $R$ could be a finite set of any positive integers larger than 1 and the number $v$ could be odd too. Under this new conditions, we are ready to present two examples as follows before entering the next section.

This seems a trivial example of a pipe of order 2: $$11 = 2 + 3^2 = 3 + 2^3$$ Only 8 pipes of order 2 of this kind are found so far. They are listed in section for simple pipe of order 2.

This example is a pipe of order 4 with even number $v$: $$22118 = 2 + 6 \cdot 3686 = 6 + 2^4 \cdot 1382 = 1382 + 2^4\cdot 6^4 = 3686 + 2^9\cdot 6^2$$ Reader should be able to verify the equations easily with or without a calculator.