“Turing was able to show that there are certain classes of problem that do not have any algorithmic solution (in particular the 'halting problem' that I shall describe shortly). However, Hilbert's actual tenth problem had to wait until 1970 before the Russian mathematician Yuri Matiyasevich-providing proofs that completed certain arguments that had been earlier put forward by the Americans Julia Robinson, Martin Davis, and Hilary Putnam-showed that there can be no computer program (algorithm) which decides yes/no systematically to the question of whether a system of Diophantine equations has a solution. It may be remarked that whenever the answer happens to be 'yes', then that fact can, in principle, be ascertained by the particular computer program that just slavishly tries all sets of integers one after the other. It is the answer 'no', on the other hand, that eludes any systematic treatment. Various sets of rules for correctly giving the answer 'no' can be provided-like the argument using even and odd numbers that rules out solutions to the second system given above-but Matisyasevich's theorem showed that these can never be exhaustive.”