Hilbert's tenth problem

Hilbert's tenth problem asks whether an algorithm can decide, for every polynomial equation with integer coefficients, whether it has an integer solution. No such algorithm exists.

Diophantine equation: A polynomial equation with integer coefficients whose solutions are sought among integers. These are precisely the equations whose integer solvability the problem asks an algorithm to decide.
David Hilbert: A German mathematician whose work shaped several major areas of mathematics in the late nineteenth and early twentieth centuries. He posed the tenth problem in his 1900 address on unsolved mathematical questions.
Martin Davis: An American mathematician whose work helped establish the link between computation and Diophantine equations. His 1950s work launched the sequence of results leading to undecidability.
Algorithm: A finite, precisely specified procedure for carrying out a computation or solving a problem. The problem asks whether one algorithm can decide every polynomial equation's integer solvability.
Matiyasevich's theorem: A theorem completing the proof that every computably enumerable set is Diophantine. It supplied the final step showing that no algorithm can solve every instance of the problem.
Hilbert's problems: A list of 23 mathematical problems presented by David Hilbert in 1900. The tenth problem was one entry in this agenda for twentieth-century mathematics.
Hilary Putnam: An American philosopher and mathematician who contributed to logic, mathematics, and philosophy of mind. He joined Davis and Julia Robinson in proving the exponential Diophantine representation result.
Diophantine equation solvability: The question of whether a specified polynomial equation has an integer solution. Individual instances can be solvable or unsolvable even though no universal decision algorithm exists.
Diophantine representation: A representation of a set of natural numbers as the parameter values satisfying a polynomial equation over the integers. Such representations let equations encode membership in computationally defined sets.
International Congress of Mathematicians: A global mathematics congress held periodically since 1897. Hilbert presented his famous list of problems at its 1900 meeting in Paris.

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