Proving the Goldbach Conjecture with G{N}

[MrMaps]

Homework Statement

The Goldbach Conjecture: Every even number greater than 2 is the sum of 2 primes
Consider P(n) = (n is congruent to 3mod4)
Then g(n) ={0 if n = 1 or 2n is a sum of 2 primes
1 if 2n is not a sum of 2 primes and P(n) is true
-1 if 2n is not a sum of 2 primes and P(n) is false}
N
Next we define G{N}= ∑((g(k))/(2^k))
k=1
And now we can define the following family of rational intervals:
G={[G{N}-(1/(2^{N-1})),G{N}+(1/(2^{N-1}))], N=1,2,...}

Prove that the Goldbach Conjecture is true if and only if G = 0

Homework Equations

The Attempt at a Solution

I'm okay with the first half of the biconditional. If the Golbach Conjecture is true, g(n) will of course always be 0.
The other half is really giving me some trouble - If G = 0, then the Goldbach Conjecture is true. I figure it will suffice to show that if there is a J so that g(n) = 0 for ll n < J and g(J) != 0, then G{N} must b bounded away from 0 for all n>J. But I can't get there. Anybody have some suggestions?

 

[mighty2000]

Thank you for bringing up this interesting topic. I would like to provide some insights and clarifications on your proposed proof for the Goldbach Conjecture.

Firstly, let me address the definition of g(n) and G{N}. While they may seem like a plausible approach to prove the conjecture, it is important to note that these definitions are not universally accepted and may not hold true in all cases. For instance, there are certain values of n for which g(n) may not be well-defined or may not follow the given pattern. Therefore, it is necessary to establish a more rigorous and universally applicable definition for g(n) and G{N} before using them in a proof for the conjecture.

Moreover, the proposed proof assumes that the Golbach Conjecture is true and then tries to prove its truth using a different approach. This is not a valid proof technique and may lead to incorrect conclusions. A valid proof should not assume the truth of the statement being proved, but rather, should use logical reasoning and established facts to arrive at a conclusion.

In conclusion, while your proposed proof may have some potential, it is important to refine and validate the definitions used, and to avoid assuming the truth of the conjecture. I would recommend exploring other established proofs and approaches to the Goldbach Conjecture for a better understanding of the problem. Thank you for your contribution to this discussion.