LAUSR

Set $${T=N^{\frac{1}{3}-\epsilon}}$$T=N13-ϵ. It is proved that for all but $${\ll TL^{-H},\,H > 0}$$≪TL-H,H>0, exceptional prime numbers $${k\leq T}$$k≤T and almost all integers b1, b2 co-prime to k, almost all integers $${n\sim N}$$n∼N satisfying $${n\equiv b_{1}+b_{2}(mod\,k)}$$n≡b1+b2(modk) can be written as the sum of two primes p1 and p2 satisfying $${p_{i}\equiv b_{i}(mod\,k),\,i=1,2}$$pi≡bi(modk),i=1,2. For prime numbers $${k\leq N^{\frac{5}{24}-\epsilon}}$$k≤N524-ϵ, this result is even true for all but $${\ll (\log\,N)^{D}}$$≪(logN)D primes k and all integers b1, b2 co-prime to k.