Geodesic
The solvability problem for equations in one unknown in nilpotent groups
Izvestiya. Mathematics, Tome 25 (1985) no. 3, pp. 601-618 
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/
@article{IM2_1985_25_3_a9,
     author = {N. N. Repin},
     title = {The solvability problem for equations in one unknown in nilpotent groups},
     journal = {Izvestiya. Mathematics},
     pages = {601--618},
     year = {1985},
     volume = {25},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1985_25_3_a9/}
}
TY  - JOUR
AU  - N. N. Repin
TI  - The solvability problem for equations in one unknown in nilpotent groups
JO  - Izvestiya. Mathematics
PY  - 1985
SP  - 601
EP  - 618
VL  - 25
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/IM2_1985_25_3_a9/
LA  - en
ID  - IM2_1985_25_3_a9
ER  - 
%0 Journal Article
%A N. N. Repin
%T The solvability problem for equations in one unknown in nilpotent groups
%J Izvestiya. Mathematics
%D 1985
%P 601-618
%V 25
%N 3
%U http://geodesic.mathdoc.fr/item/IM2_1985_25_3_a9/
%G en
%F IM2_1985_25_3_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Romankov V. L., “O nerazreshimosti problemy endomorfnoi svodimosti v svobodnykh nilpotentnykh gruppakh i v svobodnykh koltsakh”, Algebra i logika, 16:4 (1977), 457–471 | MR

[2] Repin N. N., “Uravneniya s odnoi neizvestnoi v nilpotentnykh gruppakh”, Matem. zametki, 34:2 (1983), 201–206 | MR

[3] Kholl M., Teoriya grupp, IL, M., 1962

[4] Matiyasevich Yu. V., “Diofantovy mnozhestva”, Uspekhi matem. nauk, 27:5 (1972), 185–222 | MR | Zbl

[5] Magnus V., Karras L., Soliter D., Kombinatornaya teoriya grupp, Nauka, M., 1974 | MR | Zbl

[6] Jones J. P., “Universal diophantine equations”, The Jour. of Symb. Log., 47:3 (1982), 549–571 | DOI | MR | Zbl