This question is not directly related to, but was inspired by, [this question](https://mathoverflow.net/questions/77583/is-the-free-product-of-arbitrarily-many-copies-of-mathbbz-and-mathbb). We know that a finitely generated residually nilpotent group is residually of prime-power order. However, we may need to use different primes for different elements. Classes of groups for which residual nilpotence forces there to be a single prime that will do for all elements (i.e., for which the group in question must be residually $p$-finite, for some $p$) seem to be interesting, and include, for instance, free products of cyclic groups.


**Is there a (non-cyclic) one-relator group that is residually nilpotent, but is not residually a finite $p$-group, for any prime number $p$?**


Such a group must be torsion-free, with trivial centre.



Is there a residually nilpotent one-relator group that is not residually a finite p-group for any prime p?
The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.


In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.


(Note that OP's question was posed by McCarron [2] in 1996).


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[1] *Moldavanskii, D. I.*, [**Residual nilpotence of groups with one defining relation**](https://doi.org/10.1134/S0001434620050090), translation from Mat. Zametki 107, No. 5, 752-759 (2020). [ZBL07215590](https://zbmath.org/?q=an:07215590).


[2] *McCarron, James*, [**Residually nilpotent one-relator groups with non-trivial centre**](https://www.ams.org/journals/proc/1996-124-01/S0002-9939-96-03148-6/S0002-9939-96-03148-6.pdf), Proc. Amer. Math. Soc. 124, No. 1 (1996). 


