= Diophantine equation
{wiki}
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 $(\sqrt3,0)$.
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.
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