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 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.

Proofs: 1

Proofs: 1 2
Solutions: 3

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  1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927