Legendre's equation

In mathematics, Legendre's equation is the Diophantine equation

$$ax^2+by^2+cz^2=0.$$

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