A Strengthened Binary Goldbach Theorem

Abstract

Abstract. This paper establishes a strengthened binary Goldbach theorem for every sufficiently large even integers , there exist primes and lying in the interval . The classical binary Goldbach conjecture for all sufficiently large even follows immediately as a corollary. This result yields a substantially improved upper bound for the exceptional set in Goldbach’s problem. Our argument refines the Hardy–Littlewood circle method, taking truncation parameter together with the major-arc neighborhood bound . Major arcs are treated via segmentation arguments, while minor arcs are estimated by means of exponential-sum techniques. A complete rigorous error‑term analysis is carried out throughout the main body; an appendix provides heuristic motivation for the framework.