问题

求 Jacobi 符号 $\bigl(\frac{853}{1409}\bigr)$

>> Jacobi(853,1409,show)
in> Jacobi(853,1409,show)

(556|853) = (139|853)

(6|19) = (2|19)(3|19)
(2|19) = -1

(3|19) = -(19|3) = -(1|3) = -1
(19|139) = -(139|19) = -(6|19) = -1
(139|853) = (853|139) = (19|139) = -1
(853|1409) = (1409|853) = (556|853) = -1
-------
result> -1

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从上面的结果, 可以整理得

(853|1409) = (1409|853) = (556|853)= (139|853) = (853|139) = (19|139) =  -(139|19) = -(6|19)

= -(2|19)(3|19) = -(-1)(3|19) = (3|19) = -(19|3) = -(1|3) = -1

即

\[
\begin{split}
\bigl(\frac{853}{1409}\bigr) &= \bigl(\frac{1409}{853}\bigr) = \bigl(\frac{556}{853}\bigr) = \bigl(\frac{139}{853}\bigr) = \bigl(\frac{853}{139}\bigr) = \bigl(\frac{19}{139}\bigr) =  -\bigl(\frac{139}{19}\bigr) = -\bigl(\frac{6}{19}\bigr)\\
&= -\bigl(\frac{2}{19}\bigr)\bigl(\frac{3}{19}\bigr) = -(-1)\bigl(\frac{3}{19}\bigr) = \bigl(\frac{3}{19}\bigr) = -\bigl(\frac{19}{3}\bigr) = -\bigl(\frac{1}{3}\bigr) = -1
\end{split}
\]