This law conjectured by Leonhard Euler (1707 – 1783) and first proven by Carl Friedrich Gauss (1777 – 1855).

**Theorem**: Quadratic Reciprocity Law

Let $p > 2$ and $q > 2$ be odd and distinct prime numbers. Then the the product of the Legendre symbols has the following explicit formula:

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

In particular:

- If $p\equiv q\equiv 1 \mod 4,$ then the congruences $x^2(q)\equiv p(q),$ and $x^2(p)\equiv q(p)$ are either both solvable or both not solvable.
- If $p\equiv q\equiv 3\mod 4,$ then one of the congruences $x^2(q)\equiv p(q),$ and $x^2(p)\equiv q(p)$ is solvable, the other not solvable.

| | | | | created: 2019-05-26 07:31:42 | modified: 2019-05-26 07:52:48 | by: *bookofproofs* | references: [1272]

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

(none)

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

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