(** (Order b q p) means the order of b is q in multgroup(Z_p). *)

Definition Order (b : Z) (q p : nat) :=
  0 < q /\
  Mod (Exp b q) 1 p /\ (forall d : nat, 0 < d -> Mod (Exp b d) 1 p -> q <= d).

Lemma order_ex : forall (b : Z) (p : nat), Prime p -> ~ Mod b 0 p -> exists x : nat, x < p /\ Order b x p.