Theorem (1+1.75) on the Goldbach Conjecture

Abstract

For fixed $1\le a\le2$, let Proposition $(1+a)$ assert that every sufficiently large even integer $N$ can be written as \[ N=p+rq,\qquad r\le q^{a-1}, \] where $p,q$ are primes and $r$ is either $1$ or a prime. At $a=1$, this is the binary Goldbach conjecture for sufficiently large even integers; at $a=2$, it is Chen's theorem. We prove unconditionally that every sufficiently large even integer admits such a representation with $a=1.75$, establishing Proposition $(1+1.75)$ and strengthening the earlier $(1+1.9)$ theorem of Li and Liu. The proof subtracts a positive upper-sieve estimate for excluded semiprimes from a Chen-count lower bound obtained by restricting paired terms in an eleven-term Chen weight and applying the double sieve. The threshold for $N$ is ineffective: the proof does not provide an explicit computable threshold.

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