A Logarithmic Adaptive Algorithm for Constructing Goldbach Prime Pairs in Very Large Even Integers

Abstract

Bouchaib Bahbouhi

The constructive search for Goldbach representations of very large even integers becomes increasingly difficult when direct primality testing is applied indiscriminately to candidate pairs. This work presents an adaptive computational framework designed to reduce that cost by combining centered logarithmic windows, peripheral searches, deep modular sieving, survivor ranking, adaptive zone switching, and strong probable-prime testing. For an even integer E, the algorithm uses both the centered parametrization p = E/2 − d, q = E/2 + d and the peripheral relation q = E − p. Three logarithmic scales, H2 = (log E)^2, H3 = (log E)^3, and H∞ = (log E)^3 log log E, are used as search and diagnostic scales rather than as proved universal Goldbach bounds.

The central principle of the method is MAX-SIEVE FIRST: candidates are filtered at the deepest practical sieve level before expensive primality tests are applied. Surviving candidates are ranked by resistance to progressively deeper sieving, and the algorithm dynamically switches between central, square-root, and peripheral regions when long composite-composite chains or local deserts are detected. A diagnostic SURGERY mode is introduced for difficult cases in order to distinguish low admissible density from failures of simultaneous primality synchronization.

The method was tested on even integers containing from hundreds to several thousand decimal digits. Prime-prime and probable-prime Goldbach representations were obtained in both centered and peripheral regimes, including certified examples around 10^3000 and a peripheral representation for E = 10^6000. The experiments indicate that the algorithm remains computationally efficient up to approximately 10^2000 digits, begins to slow significantly in the 10^2500– 10^3000 range, and becomes increasingly dominated by large-integer primality testing beyond approximately 10^3000 digits. These thresholds are empirical rather than mathematical limits. The results support a computational distinction between modular admissibility and bilateral prime synchronization and provide a practical framework for large-scale Goldbach pair construction.

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