A Differential Equation Approach to the Goldbach Conjecture Using the Operation and the Extended Number

This work presents a novel approach to proving the Goldbach Conjecture using nonlinear differential equations and a newly introduced mathematical operation , defined as . We define the function , which counts the number of ways an even integer can be written as the sum of two prime numbers, and derive the governing differential equation: \frac{d^2G}{dN^2} + G^2 + \left( \frac{dG}{dN} \right)^2 = a^2 + (Na + b)^2 + \frac{c}{\ln^3 N}. This equation ensures that for all even , thereby supporting the Goldbach Conjecture. The study introduces the extended number , which exhibits superposition-like properties, and demonstrates how the operation naturally arises in the analysis. Numerical verification and asymptotic analysis confirm the validity of the proposed model. The results suggest a new perspective in number theory, and further independent verification is encouraged.