View source for Legendre theorem

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==Number theory==
1) The indeterminate (Diophantine) equation

$$ax^2+by^2+cz^2=0,$$

whose coefficients $a$, $b$ and $c$ are pairwise coprime integers, square-free and not all of the same sign, has a non-zero rational solution if and only if all following three [[congruence]]s are solvable:

$$u^2\equiv-bc\pmod{|a|},$$

$$v^2\equiv-ca\pmod{|b|},$$

$$w^2\equiv-ab\pmod{|c|},$$

The question of representing the zeros of an arbitrary ternary [[quadratic form]] with rational coefficients reduces to Legendre's theorem.

It was proved by A.M. Legendre in 1785.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) pp. Chapt. 1. Par. 7 (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}} </TD></TR>
</table>



====Comments====
Legendre's theorem is an essential part of the Hasse–Minkowski theorem on rational quadratic forms (cf. [[Quadratic form|Quadratic form]]).

==Geometry==
2) The sum of the angles of a triangle cannot exceed two right angles.

3) If the sum of the angles of one triangle is equal to two right angles, then the sum of the angles of any other triangle is equal to two right angles.

Theorems 2) and 3) were proved by A.M. Legendre in 1800 and 1833 in attempts to justify Euclid's parallelism postulate (cf. [[Fifth postulate]]). Similar assertions were established by G. Saccheri (see [[Saccheri quadrangle]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> V.F. Kagan, "Foundations of geometry" , '''1''' , Moscow-Leningrad (1949) (In Russian) {{MR|0040672}} {{ZBL|0049.38001}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Pogorelov, "Lectures on the foundations of geometry" , Noordhoff (1966) (Translated from Russian) {{MR|0203551}} {{ZBL|0141.36702}} </TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian) {{MR|0120537}} {{ZBL|0211.23702}} {{ZBL|0101.37601}} {{ZBL|0052.37103}} {{ZBL|0036.10002}} {{ZBL|0061.31610}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A.P. Norden, "Elementare Einführung in die Lobatschewskische Geometrie" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) {{MR|0094759}} {{ZBL|0079.36403}} </TD></TR>
</table>

''A.B. Ivanov''