Small gaps between primes that satisfy the Goldbach equation

We study the distribution of prime numbers that satisfy the Goldbach equation. In particular, we discuss the small gaps between such prime numbers using E. Bombieri and H. Davenport’s method. We prove that, for almost all even integer N, there exist primes \(p, p'\) $p,p^{\prime }$ which \(N-p, N-p'\) $N-p,N-p^{\prime }$ are also prime and \(|p-p'| < 0.8201... \mathfrak {S}(N)^{-1}(\log N)^2\) $|p-p^{\prime }{|<0.8201...\mathfrak{S}(N)}^{-1}{(logN)}^2$ where \(\mathfrak {S}(N)\) $\mathfrak{S}(N)$ denotes the singular series for the Goldbach conjecture.