Geodesic
A~diophantine representation of perfect numbers
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VIII, Tome 88 (1979), pp. 78-89

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A natural number ft is perfect iff system (2)–(33) of diophantine equations has a positive integer solution in the rest of variables. 
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     title = {A~diophantine representation of perfect numbers},
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V. A. Kriaučiukas. A~diophantine representation of perfect numbers. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VIII, Tome 88 (1979), pp. 78-89. http://geodesic.mathdoc.fr/item/ZNSL_1979_88_a6/