On the level of distribution of Goldbach primes and its applications

We prove that, for almost all even integers $N>0$, the set of Goldbach primes $\mathbb{P} \cap (N-\mathbb{P})$ has a level of distribution $1/6$.

As applications, we show that almost all even integers $N>0$ can be written as the sum of two primes $p_1, p_2$ such that $p_1-p_2+1 \in \mathbb{P}_4$.

We also prove an analogous result with $2p_1 p_2+1 \in \mathbb{P}_{13}$ for almost all integers $N>0$ with $6\mid N$.