In 1961 Martin Davis, Hilary Putnam and Julia Robinson  proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the form
Here P is a polynomial with integer coefficients and the variables range over positive integers.
In 1970 Ju. V. Matijasevič used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevič proved  that the exponential relation y = 2x is diophantine This together with  implies that every recursively enumerable set is diophantine, i.e. every r.e. set Wcan be represented in the form
From this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets.
Now it is well known that the recursively enumerable sets W1, W2, W3, … can be enumerated in such a way that the binary relation x ∈ Wv is also recursively enumerable. Thus Matijasevič's theorem implies the existence of a diophantine equation U such that for all x and v,
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