symbol Legendre'a

[21mat]

Oblicz symbol Legendre'a korzystając z prawa wzajemności reszt kwadratowych:
\(\displaystyle{ ( \frac{71}{73})}\)

[Vax]

\(\displaystyle{ \left(\frac{71}{73}\right) = \left(\frac{73}{71}\right) = \left(\frac{2}{71}\right) = (-1)^{\frac{71^2-1}{8}} = (-1)^{\frac{70\cdot 72}{8}} = (-1)^{35\cdot 18} = 1}\)

[21mat]

Dzięki bardzo -- 5 maja 2012, o 17:34 --A jak mam \(\displaystyle{ ( \frac{-219}{383} )}\) ?

[Vax]

\(\displaystyle{ \left(\frac{-219}{383}\right) = \left(\frac{383-219}{383}\right) = \left(\frac{164}{383}\right) = \left(\frac{4}{383}\right)\left(\frac{41}{383}\right) = \left(\frac{41}{383}\right) = \left(\frac{383}{41}\right) = \left(\frac{14}{41}\right) = \left(\frac{2}{41}\right)\left(\frac{7}{41}\right) = (-1)^{\frac{41^2-1}{8}}\left(\frac{41}{7}\right) = (-1)^{\frac{40\cdot 42}{8}}\left(\frac{6}{7}\right) = (-1)^{5\cdot 42}\cdot (-1) = -1}\)