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 ${\displaystyle {}\left({\frac {53}{311}}\right)}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}\left({\frac {311}{53}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}\left({\frac {46}{53}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}\left({\frac {2}{53}}\right)\left({\frac {23}{53}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}-\left({\frac {23}{53}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}-\left({\frac {53}{23}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}-\left({\frac {7}{23}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}\left({\frac {23}{7}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}\left({\frac {2}{7}}\right)}$ ${\displaystyle {}\,\,}$ ${\displaystyle {}\,=\,}$ ${\displaystyle {}1}$
Also ist ${\displaystyle {}53}$ ein Quadratrest modulo ${\displaystyle {}311}$.