 For a positive solution of Hilbert’s tenth problem, it would have sufficed to present a particular concrete method which would have intuitively been a “mechanical” decision method. However, Turing’s pioneering analysis of the notion of decision method brought into focus the possibility of a negative solution. Beginning in the early 1950s, Julia Robinson and Martin Davis worked on this problem, later joined by Hilary Putnam. As a result of their collaboration, the first important result in this direction was achieved. Call an equation “an exponential Diophantine equation” if it involves also exponentiation, as well as addition and multiplication (that is, one can have both constants and variables as exponents); naturally, the focus is still in the integer solutions. Davis, Putnam, and Robinson (1961), showed that the problem of solvability of exponential Diophantine equations is undecidable. In 1970, Yuri Matiyasevich added the final missing piece, and demonstrated that the problem of the solvability of Diophantine equations is undecidable. Hence the overall result is often called MRDP Theorem (for an exposition, see, e.g., Davis 1973; Matiyasevich 1993).