Quadratic Reciprocity Theorem

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Quadratic reciprocity is a classic result of number theory.
It is one of the most important theorems in the study of quadratic residues.


It states that $\left(\frac{p}{q}\right)= \left(\frac{q}{p}\right)$ for primes $p$ and $q$ greater than $2$ where both are not of the form $4n+3$ for some integer $n$.
If both $p$ and $q$ are of the form $4n+3$, then $\left(\frac{p}{q}\right)= -\left(\frac{q}{p}\right).$

Another way to state this is:

  • Note that $\left(\frac{p}{q}\right)$ is not a fraction. It is the Legendre notation of quadratic residuary.

See Also

Quadratic Residues