The $$R_\infty $$ property for pure Artin braid groups

In this paper we prove that all pure Artin braid groups $$P_n$$ ( $$n\ge 3$$ ) have the $$R_\infty $$ property. order to obtain result, analyse naturally induced morphism $${\text {Aut}}\left( {P_n}\right) \longrightarrow {\text {\Gamma _2 (P_n)/\Gamma _3(P_n)}\right) which turns out factor through a representation $$\rho :S_{{n+1}} . We can then use theory of symmetric show any automorphism $$\alpha acts on free abelian group $$\Gamma _3(P_n)$$ via matrix with an eigenvalue equal 1. This allows us conclude Reidemeister number $$R(\alpha )$$ is $$\infty