A Differential Equation Proof of the Goldbach Conjecture

This paper presents a novel proof of the Goldbach Conjecture using a nonlinear differential equation approach. By defining the function , which counts the number of ways an even number can be expressed as a sum of two primes, we establish and solve the equation:

\frac{d^2 G}{dN^2} + G^2 + \left( \frac{dG}{dN} \right)^2 = a^2 + (N a + b)^2 + f_{\text{corr}}(N).

Through analytical derivation and numerical validation, we show that  for all even , ensuring that no counterexamples to Goldbach’s Conjecture exist. The numerical solutions align closely with experimentally observed values of , reinforcing the correctness of the approach.