User blog comment:Vel!/Call for math facts/@comment-5529393-20140731034954/@comment-1605058-20140801105240

Turns out that for the second fact we need generalized Elliot-Halberstam conjecture. The best result so far using EH alone is D=12. It can also be noted (possibly in separate fact) that we can't get better than D=6 using sieve methods, because so called "parity obstructions".