The Goldbach's conjecture is that $\forall n \in \mathbb{N}^*, \ \exists p,q \in \mathcal{P}$ such that $2n=p+q$.

I wonder if there are some generalization giving more information about the congruence satisfied by $p,q$, for example

Given $m ,a\in \mathbb{N},gcd(a,m)=1$, for every $ n \equiv a \pmod{m}$ large enough, $\exists p,q \in \mathcal{P}$ such that $2n=p+q$ and $p\equiv q \equiv a \pmod{m}$

Is there some reference on this, and is it much harder than the original Goldbach's conjecture ?


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