Legendre's equation

In mathematics, Legendre's equation is a Diophantine equation of the form:

$${\displaystyle ax^{2}+by^{2}+cz^{2}=0.}$$

The equation is named for Adrien-Marie Legendre who proved it in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers and also not all positive or all negative.