Abstract: World Journal of Applied Mathematics and Statistics

Abstract:
We present a methodology for constructing explicit Goldbach decompositions E = p + q for extremely large even integers E, based on a narrow central logarithmic window around E/2 and a residue lane filtering strategy. The method combines modular constraints, partial sieving, and probabilistic primality testing to produce concrete examples rather than exhaustive verification. Using this approach, we report certified or high confidence examples up to E = 10^1300. The emphasis of this work is methodological: we describe a structured and scalable workflow that drastically reduces the search space while remaining compatible with standard primality tests such as Miller–Rabin and elliptic curve primality proving. The results are explicit pointwise constructions and do not constitute a full verification of Goldbach’s conjecture up to the stated bounds.