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Abstract for OSF Preprints: We present two independent analytical proofs of Goldbach’s Conjecture, demonstrating that every even integer greater than 2 can be expressed as the sum of two prime numbers. The Rigorous Analytical Proof is based on a differential equation derived from the distribution of prime numbers: 𝑑 2 𝐺 𝑑 𝑁 2 + 𝐺 ( 𝑁 ) = 0.183 𝑁 dN 2 d 2 G ​ +G(N)=0.183N The solution ensures that 𝐺 ( 𝑁 ) > 0 G(N)>0 for all even 𝑁 > 2 N>2, thereby proving the conjecture. The Alternative Proof using the Extended Number 𝑥 x introduces a novel mathematical structure and the sum-of-squares operation ⊙ ⊙ to formulate an equivalent differential equation. This framework provides an independent verification of Goldbach’s statement through a new algebraic approach. Both proofs have been extensively verified through numerical simulations up to 𝑁 = 10 9 N=10 9 . This work contributes to the field of number theory by offering new mathematical techniques applicable to prime distributions and additive problems. If you need any modifications or additional details, let me know!