 This conjecture was proven for all formula_20 by Enrico Bombieri and A. I. Vinogradov (the Bombieri–Vinogradov theorem, sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the generalized Riemann hypothesis.It is known that the conjecture fails at the endpoint formula_21.The Elliott–Halberstam conjecture has several consequences.