Geodesic
Diophantine representation of enumerable predicates
Izvestiya. Mathematics , Tome 5 (1971) no. 1, pp. 1-28

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An example is given of a diophantine relation which has exponential growth. This, together with the well-known results of Martin Davis, Hilary Putnam, and Julia Robinson, yields a proof that every enumerable predicate is Diophantine. This theorem implies that Hilbert's tenth problem is algorithmically unsolvable. 
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Yu. V. Matiyasevich. Diophantine representation of enumerable predicates. Izvestiya. Mathematics , Tome 5 (1971) no. 1, pp. 1-28. http://geodesic.mathdoc.fr/item/IM2_1971_5_1_a0/