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## Definition: Legendre Symbol

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:
$$\left(\frac np\right):=\begin{cases} 1&\text{if }n\text{ is quadratic residue modulo }p\text{ and }p\not\mid n,\\ -1&\text{if }n\text{ is a quadratic nonresidue modulo }p\text{ and }p\not\mid n,\\ 0&\text{if }p\mid n. \end{cases}$$

### Examples

• 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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### Bibliography (further reading)

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