Geodesic
On Diophantine representations of the sequence of solutions of Pell's equation
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 49-59

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The paper deals with construction of sufficiently simple Diophantine representations of the sequence of Pell's equation solutions. Such a representation is transformed into a Diophantine representation of the predicate $x=y^z$. It is also proved that there exists a universal polynomial of some simple form. 
@article{ZNSL_1971_20_a5,
     author = {N. K. Kossovski},
     title = {On {Diophantine} representations of the sequence of solutions of {Pell's} equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {49--59},
     publisher = {mathdoc},
     volume = {20},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a5/}
}
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N. K. Kossovski. On Diophantine representations of the sequence of solutions of Pell's equation. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 49-59. http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a5/