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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.