 Diophantine equation

A Diophantine equation is a polynomial equation for which integer solutions are sought. Here the polynomial coefficients are also integers. A real solution alone need not meet this arithmetic requirement.

For example, $x^2+y^2=z^2$ has the Pythagorean triple $(3,4,5)$ as a solution. In contrast, $x^2+y^2=3$ has no integer solution: a perfect square is zero or one modulo four, so their sum cannot be three. The circle still has real points, such as $(\sqrt{3},0)$.

A linear Diophantine equation is controlled by a greatest common divisor and Bézout's identity. Higher-degree equations can require integer factorization, quadratic reciprocity, infinite descent or bounds on arithmetic height. A primitive Diophantine solution separates common scaling when the equation is defined by a homogeneous polynomial, while local solubility tests the equation over completions before attempting a global solution.

A Diophantine equation is a polynomial equation of the form: \[ P(x_1, x_2, \ldots, x_n) = 0 \] where \( P \) is a polynomial with integer coefficients, and the solutions \( (x_1, x_2, \ldots, x_n) \) are required to be integers.

Polynomial (possibly a multivariate polynomial) with integer coefficients.

Sometimes systems of Diophantine equations are considered.

Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.

The general problem is known to be undecidable: Hilbert's tenth problem.

The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.