**Theorem**: Second Supplementary Law to the Quadratic Reciprocity Law

For a prime number $p > 2$ the following formula for the Legendre symbol holds:

$$\left(\frac {2}p\right)=(-1)^{\frac{p^2-1}{8}}.$$

More in detail, this law states that

$$\left(\frac {2}p\right)=\begin{cases}1&\text{if }p\equiv \pm 1\mod 8,\\-1&\text{if }p\equiv \pm 3\mod 8.\end{cases}$$

In particular, the congruence $x^2(p)\equiv 2(p)$ is only solvable, if $p$ has the form $p\equiv \pm 1\mod 8,$ and any odd prime factor of the integer $x^2-2$ has the form $p\equiv \pm 1\mod 8.$

| | | | | created: 2019-05-26 07:31:42 | modified: 2019-05-26 18:18:17 | by: *bookofproofs* | references: [1272]

## 1.**Proof**: *(related to "Second Supplementary Law to the Quadratic Reciprocity Law")*

(none)

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

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