Exercise 3.3.1 (Some Jacobi symbols)

Question

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}
Evaluate: $\legendre{-23}{83}; \legendre{51}{71}; \legendre{71}{73}; \legendre{-35}{97}$.

Richard Ganaye · Accepted Answer

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}

Using Theorems 3.6, 3.7, 3.8, we obtain
\begin{enumerate}
\item[(a)] 
\begin{align*}
\legendre{-23}{83} & =\legendre{-1}{83} \legendre{23}{83}& (83 \equiv -1 \pmod 4)\ 
&= - \legendre{23}{83} = \legendre{83}{23} & (23 \equiv -1 \pmod 4)\
&= \legendre{14}{23}
= \legendre{2}{23} \legendre{7}{23} & (23 \equiv -1 \pmod 8)\
&=  \legendre{7}{23} = -\legendre{23}{7} & (\phantom{1}7 \equiv -1 \pmod 4)\
&= -\legendre{2}{7}= -1.
\end{align*}

\item[(b)] Similarly
\begin{align*}
&\legendre{51}{71} = - \legendre{71}{51} = -\legendre{20}{51} = - \legendre{2}{51}^2 \legendre{5}{51} = -\legendre{5}{51} = - \legendre{51}{5} = - \legendre{1}{5} = -1.
\end{align*} 
 \item[(c)] 
 \begin{align*}
 &\legendre{71}{73} = \legendre{-2}{73} = \legendre{-1}{73} \legendre{2}{73} = 1.
 \end{align*}
 
 \item[(d)] 
 \begin{align*}
 &\legendre{-35}{97} = \legendre{-1}{97}\legendre{35}{97} = \legendre{35}{97} = \legendre{97}{35} = \legendre{27}{35} = -\legendre{35}{27} = -\legendre{8}{27} = - \legendre{2}{27}^3 = 1.
 \end{align*}
\end{enumerate}
\end{proof}
{\bf Check:}
\begin{verbatim}
sage: j = kronecker
sage: j(-23,83), j(51,71), j(71,73), j(-35,97)
(-1, -1, 1, 1)
\end{verbatim}

Exercise 3.3.1 (Some Jacobi symbols)

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Exercise 3.3.1 (Some Jacobi symbols)

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