On the Goldbach conjecture in arithmetic progressions

It is proved that for a given integer N and for all but << (log N)(B) prime numbers k <= N5/48-epsilon the following is true: For any positive integers b(i), i is an element of {1, 2, 3} (b(i), k) = 1 that satisfy N equivalent to b(1) + b(2) + b(3) (mod k), N can be written as N = p(1) + p(2) + p(3), where the p(i), i is an element of {1, 2, 3} are prime numbers that satisfy p(i) b(i) (mod k).