Theorem of Legendre

The theorem of Legendre which follows relates to the equations diophantiennes form $\, ax^2 + by^2 + cz^2 = 0$ where the coefficients $\, has, B, c$ satisfy the following assumptions:

(I) $\, has > 0$ , $\, B < 0$ and $\, C < 0$ ,

(II) $\, has, B, c$ is without square factor and first between them two to two.

The theorem of Legendre stipulates whereas the equation diophantienne above has a solution (noncommonplace) if and only if:

$\, - ab$ is quadratic Résidu $\, \ pmod c$ ,

$\, - bc$ is quadratic residue $\, \ pmod a$

and

$\, - ca$ is quadratic residue $\, \ pmod b$ ,

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