Exercise 3.1.4 (Some values of $(a/p)$)

Question

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}
Find the values of $\legendre{a}{p}$ in each of the 12 cases, $a = -1, 2, -2, 3$ and $p=11,13,17$.

Richard Ganaye · Accepted Answer

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

\begin{proof} 
Using the sage function ``kronecker'', we obtain

\begin{center}
\begin{tabular}{|c|cccc|}
\hline
 &  $-1$ &  2 & $-2$ & 3 \
\hline
11 &    -1& -1  & 1    & 1\
13 &   1 &  -1 & -1   & 1\
17 &    1&  1  & 1   & -1 \
\hline
\end{tabular}
\end{center}
\end{proof}
\begin{verbatim}
sage: for p in [11,13,17]:
....:         print(p, [kronecker(a,p) for a in [-1,2,-2,3]])
....:     
(11, [-1, -1, 1, 1])
(13, [1, -1, -1, 1])
(17, [1, 1, 1, -1])
\end{verbatim}

Exercise 3.1.4 (Some values of $(a/p)$)

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Exercise 3.1.4 (Some values of $(a/p)$)

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