Two Problems around the Parity Principle

We shall present two recent results linked with the Parity Principle. First, with Aled Walker, we proved that given any $X\ge2$ each progression $a$ mod $q$, with $a$ prime to $q$, contains a product of exactly three primes of size at most $X$ provided that $q\le X^{3/16}$; Bound improved to $q\le X^{1/3}/900$ with Priyamvad Srivastav recently. Our sole ingredients being a sieve bound and a weak lower bound for $L(1,\chi)$, this apparently contradicts the Parity Principle but no: the Parity Principle deals with quantitative estimates while our proof does not produce enough products! Secondly, if we are to keep asymptotic estimates and sieve devices, primes are accompanied with products of two primes, but how can one be more specific? Recently N. Debouzy refined an estimate of E. Bombieri and obtained, under the Elliott-Hamberstam conjecture, an asymptotic for the number of primes $p$ such that $p+2$ is either a prime or a product of two primes, one of them being of size at most $p^\epsilon$, for any arbitrary $\epsilon>0$.