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

在本文中，我们证明所有纯Artin编织组\（P_n \）（\（n \ ge 3 \））具有\（R_ \ infty \）属性。为了获得此结果，我们分析自然诱发的态射\\ {{text {Aut}} \ left（{P_n} \ right）\ longrightarrow {\ text {Aut}} \ left（{\ Gamma _2（P_n） / \ Gamma _3（P_n）} \ right）\）会通过表示形式\（\ rho：S _ {{n + 1}} \ longrightarrow {\ text {Aut}} \ left（{\ Gamma _2 （P_n）/ \ Gamma _3（P_n）} \ right）\）。然后，我们可以使用对称组的表示理论来证明\（P_n \）的任何自同构\（\ alpha \）都作用于自由阿贝尔群\（\ Gamma _2（P_n）/ \ Gamma _3（P_n）\）经由与本征值等于1。这使我们得出结论一个矩阵，该数Reidemeister \（R（\阿尔法）\）的\（\阿尔法\）是\（\ infty \） 。