Legendre's equation

In mathematics, Legendre's equation is the Diophantine equation

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

You might be looking for Legendre's differential equation.

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 −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, not all positive or all negative .

References

• L. E. Dickson, History of the Theory of Numbers. Vol.II: Diophantine Analysis, Chelsea Publishing, 1971, ISBN 0-8284-0086-5. Chap.XIII, p. 422.
• J.E. Cremona and D. Rusin, "Efficient solution of rational conics", Math. Comp., 72 (2003) pp. 1417-1441.