In mathematics, Legendre's equation is the
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
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 residue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that:
:x^2\equiv q \pmod.
Otherwise, ''q'' is called a quadratic no ...
s 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
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remem ...
, ''
History of the Theory of Numbers. Vol.II: Diophantine Analysis'',
Chelsea Publishing, 1971, . Chap.XIII, p. 422.
* J.E. Cremona and D. Rusin, "Efficient solution of rational conics",
Math. Comp., 72 (2003) pp. 1417-1441.
Diophantine equations
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