Abstract

The computational construction of Goldbach prime pairs becomes increasingly demanding when the prescribed even integer reaches several thousand decimal digits, because repeated primality testing of unsuccessful candidates can rapidly dominate the computational cost. This work introduces Bahbouhi.b.goldbach.2026, an adaptive multilevel algorithm designed to construct prime pairs p and q satisfying p + q = E for a prescribed very large even integer E. Instead of relying on a direct sequential search, the method organizes candidate selection into a hierarchy of increasingly expensive operations. It combines arithmetic classification modulo 6 and 12, admissible residue rails, destructive congruences, an r - 1 divisibility representation, deep modular sieving, survivor-resistance ranking, peripheral and central search branches, local candidate probes, anchor propagation, and an adaptive controller that redirects the search when a branch becomes unproductive. Probable-prime testing is therefore concentrated on candidates that survive the cheaper arithmetic layers. The algorithm was experimentally applied to prescribed even integers ranging from moderate sizes to more than 10,000 decimal digits. The largest externally verified representation retained in the reported experiments is E = 10^10068 + 28878, with p = 16481 and q = 10^10068 + 12397. The experiments illustrate the practical scalability of the multilevel approach while identifying the principal computational bottlenecks that emerge at extreme digit lengths. This work is computational and algorithmic and makes no claim of proving Goldbach’s conjecture.