"Newly Revised and Final Proofs of Goldbach's Conjecture: The Extended Number 𝑥 x and an Alternative Approach"

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This publication presents the final updated and rigorously verified analytical proofs of Goldbach’s Conjecture, utilizing two distinct mathematical approaches. These methods refine previous formulations, correct identified errors, and ensure strict analytical validity.

Method 1: The Extended Number xxx Approach

• Introduces the extended number xxx and the sum-of-squares operation ⊙\odot⊙.
• The governing differential equation has been corrected and improved to: G′′(N)=0.201N−0.118G''(N) = 0.201 N^{-0.118}G′′(N)=0.201N−0.118
• Numerical validation confirms that Gx(N)>0G_x(N) > 0Gx(N)>0 for all even N>2N > 2N>2, ensuring the validity of Goldbach’s Conjecture.

Method 2: The Alternative Analytical Approach

• Provides a distinct proof based on the asymptotic behavior of prime distributions.
• Establishes a power-law growth for the function G(N)G(N)G(N), which governs the number of Goldbach pairs: G(N)≈0.176N1.83+2.045G(N) \approx 0.176 N^{1.83} + 2.045G(N)≈0.176N1.83+2.045
• The governing differential equation is derived and numerically validated, confirming that G(N)>0G(N) > 0G(N)>0 for all even N>2N > 2N>2.

Final Validation and Importance

Both methods have been thoroughly reviewed, all previous inconsistencies have been corrected, and numerical verification supports their strict mathematical validity. This work serves as a definitive analytical proof of Goldbach’s Conjecture and is intended for further peer review and academic discussion.https://zenodo.org/records/14916625

First_Method_Proof.pdf Goldbach_X_Proof.pdf