Legendre's equation

For the equation satisfied by Legendre polynomials, see Legendre's differential 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 −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. [1]