Editing Hilbert's tenth problem (section)

=== Original formulation ===
Hilbert formulated the problem as follows:{{sfn|Hilbert|1902|p=458}}

<blockquote>''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 in a finite number of operations whether the equation is solvable in rational integers.''</blockquote>

The words "process" and "finite number of operations" have been taken to mean that Hilbert was asking for an [[algorithm]]. The term "rational integral" simply refers to the integers, positive, negative or zero: 0, ±1, ±2, ... .  So Hilbert was asking for a general algorithm to decide whether a given polynomial [[Diophantine equation]] with integer coefficients has a solution in integers.

Hilbert's problem is not concerned with finding the solutions. It only asks whether, in general, we can decide whether one or more solutions exist. The answer to this question is negative, in the sense that no "process can be devised" for answering that question. In modern terms, Hilbert's 10th problem is an [[undecidable problem]].