In mathematics, Legendre's equation is a Diophantine equation of the form: a x 2 + b y 2 + c z 2 = 0. {\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.
In mathematics, Legendre's equation is a Diophantine equation of the form: a x 2 + b y 2 + c z 2 = 0. {\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.
Source: Wikipedia "Legendre's equation" · CC BY-SA 4.0
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