Geodesic
The solvability problem for equations in one unknown in nilpotent groups
Izvestiya. Mathematics , Tome 25 (1985) no. 3, pp. 601-618.

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The author constructs a finitely generated nilpotent group of class 3 for which there is no algorithm recognizing the solvability of equations in one unknown. Such an algorithm exists for every finitely generated nilpotent group of class 2. It is proved that for any $c\geqslant10^{20}$ there is no algorithm recognizing the solvability of equations in one unknown in free nilpotent groups of class $c$. Bibliography: 6 titles. 
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N. N. Repin. The solvability problem for equations in one unknown in nilpotent groups. Izvestiya. Mathematics , Tome 25 (1985) no. 3, pp. 601-618. http://geodesic.mathdoc.fr/item/IM2_1985_25_3_a9/

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