A Logarithmic Window Algorithm for Constructive Goldbach Representations near the Centre of Large Even Integersc

Abstract

Goldbach's conjecture remains one of the oldest unsolved problems in number theory despite substantial theoretical and computational progress over the last three centuries. Classical computational approaches generally rely on exhaustive searches over large portions of the interval [2, E/2], making their complexity increase rapidly with the size of the even integer E. In this work, we introduce a new constructive algorithm based on a centre-deviation representation of Goldbach pairs. Instead of scanning the entire search interval, the proposed method investigates only a logarithmic neighbourhood centred at E/2. Candidate prime pairs are generated according to E = (H − t) + (H + t), where H = E/2 and t denotes the deviation from the centre. The search window is restricted to W = C(log E)^2, where C is an experimentally determined constant.
The proposed framework transforms the classical additive formulation of Goldbach's conjecture into a symmetric search problem around the centre of the even integer. Experimental computations demonstrate that the logarithmic window is sufficient to recover Goldbach representations for a wide range of tested integers, including examples exceeding 10^1300. The algorithm is combined with deterministic primality certification for moderate-size integers and ECPP certification for very large integers.
A theoretical analysis of the computational complexity is presented and compared with brute-force approaches and previously reported computational strategies. Several graphical representations illustrate the geometric interpretation of the method, the logarithmic search window, the distribution of deviations, and the computational gains obtained. Although no proof of Goldbach's conjecture is claimed, the results strongly suggest that symmetric logarithmic neighbourhoods constitute a remarkably efficient region for locating Goldbach representations.