Let $p > 2$ be an odd and fixed prime number. The **Legendre symbol** $p$ $\left(\frac np\right)$ is an arithmetic function defined using the quadratic residues modulo $p$ as follows:

$$\left(\frac np\right):=\begin{cases}

1&\text{if }n\text{ is quadratic residue modulo }p\text{ and }p\not\mid n,\\

-1&\text{if }n\text{ is a quadratic nonresidue modulo }p\text{ and }p\not\mid n,\\

0&\text{if }p\mid n.

\end{cases}$$

- If $n$ is odd and $p\not\mid m$, then $\left(\frac {n^2}p\right)=1.$
- $\left(\frac 1p\right)=1$ for all $p > 2.$

| | | | | created: 2019-05-12 09:56:34 | modified: 2019-05-15 05:35:51 | by: *bookofproofs* | references: [1272]

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[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927