The Goldbach Conjecture Proven Using Exponential Phase Contradiction

We present an elegant and elementary proof of the three-century-old Goldbach Conjecture based on a novel application of complex exponential phase identities. Assuming, for contradiction, that an even number R > 4 cannot be written as the sum of two primes, we analyze its decomposition through the lens of Euler’s identity: exp(iπR/2) = -1. When one term is prime and the other composite, the composite is shown to admit a secondary decomposition involving a prime and an even number. Tracking the resulting phase contributions reveals an internal contradiction: the parity of the prime sum conflicts with the required unit-circle rotation, leading to incompatible exponential evaluations. This contradiction eliminates the possibility of such a counterexample, thereby proving that every even integer greater than 4 is the sum of two primes.