A plunger kit for a medicine syringe provided with a syringe barrel includes a piston having an internal screw and a plunger rod having an external screw. The internal screw and external screw have a mutually non-complementary, basic groove profile and basic thread profile, respectively. A following flank of each thread of the external screw and a groove flank of a corresponding groove of the internal screw are formed such that, when the piston and plunger rod are slidably fit in the syringe barrel and the piston is then slid in a pull direction, the following flank can come into contact with the groove flank more strongly at a first part and more weakly at a second part to increase thread engagement strength between the external screw and the internal screw without any substantial expansion of the piston by the external screw.
