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## Lemma: Gaussian Lemma (Number Theory)

Let $p > 2$ be an odd prime number and let $n$ be an integer not divisible by $p$ ($p\not\mid n$). The Legendre symbol $\left(\frac np\right)$ can be calculated via the formula
$$\left(\frac np\right)=(-1)^m,$$
where $m\ge 0$ is the number of congruence classes modulo $p$ represented by $\frac{p-1}2$ residues $$n,2n,3n,\dots ,{\frac {p-1}{2}}n\mod p$$
which are $> \frac{p}{2}$ (i.e. $\ge\frac{p+1}2$).

| | | | | created: 2019-05-26 18:42:48 | modified: 2019-05-26 20:01:15 | by: bookofproofs | references: [1272]

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