  In this paper we prove that the binary Goldbach conjecture for sufficiently
large even integers would follow under the assumption that both the
Elliott-Halberstam conjecture and a variant of the Elliott-Halberstam
conjecture twisted by the M\"{o}bius function hold, in which the sum of their
levels of distribution exceeds 1. This continues the work of Pan in 1982. An
analogous result for the twin prime conjecture is obtained by Ram Murty and
Vatwani in 2017.
