Hilbert’s 10th Problem
Hilbert’s 10th problem was the only one of the 23 Paris Problems that was a decision problem. It was to devise an algorithm to determine if a given Diophantine equation was solvable. It was to specify a procedure which, in a finite number of steps, enables one to determine whether or not a given Diophantine equation with an arbitrary number of indeterminates and with integral coefficients has a solution in rational integers.
The formal statement of the 10th Problem as given by Hilbert:
10. Determination of the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
The 10th problem was proven impossible by Julia Robinson and Martin Davis in 1970.
In 1948, after receiving her doctorate at the University of California at Berkeley, Julia Robinson began her work on Hilbert’s 10th problem. This problem occupied most of her professional career. Along with Martin Davis and Hilary Putman, she gave a fundamental result which contributed to the solution to Hilbert's Tenth Problem. The Davis-Putnam-Robinson paper was presented in 1961. She worked on the problem for over twenty years, building a foundation which Yuri Matiyasevic used in 1970 to prove that there is not a general method for determining solvability. She also did important work on that problem with Yuri Matiyasevic after he gave the solution in 1970.
Thus, Hilbert's 10th problem was solved in the negative, in that there is no algorithms to determine if a given diophantine equation was solvable.
In 1930, Hilbert retired and the city of Königsberg made him an honorary citizen of the city. He gave an address which ended with six famous words showing his enthusiasm for mathematics and his life devoted to solving mathematical problems: “Wir mussen wissen, wir werden wissen” – “We must know, we shall know.”