Primes in simultaneous arithmetic progressions

We prove a new mean value theorem on the distribution of primes in two simultaneous arithmetic progressions. Our approach builds on previous arguments of Bombieri, Fouvry, Friedlander, and Iwaniec appealing to spectral theory of Kloosterman sums, as well as the $q$-analogue of van der Corput method. In particular, we need estimates for exponential sums coming from the spectral theory of automorphic forms (sums of Kloosterman sums) and from algebraic geometry (Weil--Deligne bound for algebraic exponential sums). As an application, we show that the greatest prime factor of $p + 6$ for Chen prime $p$ is infinitely often greater than $p^{0.217}$.