Residual Nilpotence of Groups with One Defining Relation

All groups in the family of Baumslag–Solitar groups (i.e., groups of the form $G(m,n)=\langle a, b; \,a^{-1}b^ma=b^n \rangle$, where $m$ and $n$ are nonzero integers) for which the residual nilpotence condition holds if and only if the residual $p$-finiteness condition holds for some prime number $p$ are described. It has turned out, in particular, that the group $G(p^r,-p^r)$, where $p$ is an odd prime and $r\ge1$, is residually nilpotent, but it is residually $q$-finite for no prime $q$. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual $p$-finite for some prime $p$, which was announced in the same paper of McCarron, is also given.