Empirical Verification of the Weak Goldbach Conjecture for Odd Integers up to 4300 Digits(≈ 5.075×〖10〗^4299 ): A Novel Computational Approach

The Weak Goldbach Conjecture asserts that every odd integer greater than 5 can be expressed as the sum of three primes. While Helfgott (2013) provided a theoretical proof, computational verification remains critical for extremely large integers. This study presents a novel computational framework that extends empirical validation to odd integers with up to 4300 digits (≈5.075×〖10〗^4299), a significant leap beyond previous limits. Our approach integrates a probabilistic primality test with an optimized partitioning algorithm, systematically exploring configurations of the form (e_1+p,e_2+p,c), where e_1,e_2 are even, p is odd, and c=O-(e_1+e_2+2p), ensuring all terms are prime. Utilizing optimized high-precision arithmetic and parallel processing techniques, we verified the conjecture for random odd integers O with 9≤d≤4300 digits, achieving runtimes under 43 minutes for the largest cases. Independent verification confirmed the primality of all triplet components, ensuring mathematical rigor. This work highlights the potential of computational methods in number theory.