Legendre Symbol of Congruent Integers
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Theorem
Let $p$ be a odd prime.
Let $a, b \in \Z$ be such that $a \equiv b \pmod p$.
Then:
- $\paren {\dfrac a p} = \paren {\dfrac b p}$
where $\paren {\dfrac a p}$ is the Legendre symbol.
Proof
\(\ds \paren {\frac a p}\) | \(=\) | \(\ds a^{\frac {p - 1} 2} \bmod p\) | Definition 2 of Legendre Symbol | |||||||||||
\(\ds \) | \(=\) | \(\ds b^{\frac {p - 1} 2} \bmod p\) | Congruence of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac b p}\) | Definition 2 of Legendre Symbol |
$\blacksquare$