Diophantine equations are equations of the form f = 0, where f is a polynomial with integer coefficients in one or several variables which take on integral values. They are named after Diophantus who studied equations with variables which take on rational values. Examples of Diophantine equations are

  • ax + by = 1: See Bézout's identity.
  • xn + yn = zn: For n=2 there are many solutions (x,y,z), the Pythagorean triples. For larger values of n, Fermat's Last Theorem states that no positive integer solutions x, y, z satisfying the above equation exist.
  • x2 - n y2 = 1: (Pell's equation) which is named, mistakenly, after the English mathematician John Pell. It was studied by Fermat.

The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable.

The field of Diophantine approximation deals with the cases of Diophantine inequalities: variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds.