Products of primes in arithmetic progressions: a footnote in parity   breaking
  We prove that, if $x$ and $q\leqslant x^{1/16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1/3}$, that is congruent to $a$ modulo $q$. 
Olivier Ramar\'e, Aled Walker
math.NT