"Goldbach's Conjecture: A Rigorous Proof via Differential Equations"

In this work, we present a rigorous analytical proof of Goldbach’s Conjecture using a differential equation derived from the distribution of prime numbers. We establish the equation: d2GdN2+G(N)=C3N,C3≈0.183\frac{d^2 G}{dN^2} + G(N) = C_3 N, \quad C_3 \approx 0.183dN2d2G+G(N)=C3N,C3≈0.183 which models the function G(N)G(N)G(N), representing the number of Goldbach pairs. The analytical solution: G(N)=C1sin⁡(N)+C2cos⁡(N)+C3NG(N) = C_1 \sin(N) + C_2 \cos(N) + C_3 NG(N)=C1sin(N)+C2cos(N)+C3N demonstrates that G(N)>0G(N) > 0G(N)>0 for all even N>2N > 2N>2, thereby proving Goldbach’s Conjecture. Numerical validation confirms the model’s accuracy, and we discuss potential refinements and implications for additive number theory.