We prove a conjecture due to Makanin: if and are elements of the Artin braid group such that for some nonzero integer , then and are conjugate. The proof involves the Nielsen–Thurston classification of braids.
"The $n$th root of a braid is unique up to conjugacy." Algebr. Geom. Topol. 3 (2) 1103 - 1118, 2003. https://doi.org/10.2140/agt.2003.3.1103