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{{dablink|You might be looking for [[Legendre polynomials|Legendre's differential equation]].}}

In mathematics, '''Legendre's equation''' is the [[Diophantine equation]]

:<math>ax^2+by^2+cz^2=0.</math>

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 residue]]s modulo ''a'', ''b'' and ''c'', respectively, where ''a'', ''b'', ''c'' are nonzero, [[Square-free integer|square-free]], [[Pairwise coprime|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", [[Mathematics of Computation|Math. Comp.]], '''72''' (2003) pp.1417-1441.  [http://www.warwick.ac.uk/staff/J.E.Cremona/papers/conics.pdf]

[[Category:Diophantine equations]]

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