Exercise 3.2.10 (Primes for which $-2$ a quadratic residue)

Question

Of which primes is $-2$ a quadratic residue?

Richard Ganaye · Accepted Answer

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

\begin{proof}  The number $-2$ is a residue modulo $p$ if and only if $\legendre{-2}{p} = 1$, where
\begin{align*}
\legendre{-2}{p} &= \legendre{-1}{p}\legendre{2}{p}\
&=(-1)^{(p-1)/2} (-1)^{(p^2 - 1)/8}.
\end{align*}
Thus $\legendre{-2}{p} = 1$ if and only if
\begin{align*}
(p \equiv 1 \pmod 4 &	ext{ and } p \equiv 1, 7 \pmod 8)\
&	ext{or}\
(p\equiv 3 \pmod 4 &	ext{ and } p \equiv 3,5 \pmod 8).
\end{align*}
which gives $p \equiv 1, 3 \pmod 8$.

$$\legendre{-2}{p} = 1 \iff p \equiv 1 \pmod 8 	ext{ or } p \equiv 3 \pmod 8.$$
(The first such primes are $3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137,\ldots$)
\end{proof}

Exercise 3.2.10 (Primes for which $-2$ a quadratic residue)

Answers

Comments

Exercise 3.2.10 (Primes for which $-2$ a quadratic residue)

Answers

Comments

Add answer