Applications of quadratic residues

[matqkks]

I have covered the proofs of the laws of quadratic reciprocity (the Legendre and Jacobi symbols). However this treatment of quadratic residues has been pretty dry. Are there any real life applications of the quadratic residues?

 

[ThePerfectHacker]

Not that I know of. If you are doing mathematics you care about its applications to other parts of mathematics, and for that there are plentiful.

Here is one application within mathematics. A Dirichlet character mod $n$ is a function $\chi:\mathbb{Z} \to \mathbb{C}$ such that $\chi(a) = \chi(a+n)$ and $\chi(ab) = \chi(a)\chi(b)$. It also has one more additional property, it only pays attention to the integers relatively prime to $n$, in other words, $\chi(a) = 0$ if $(a,n) > 0$ and $\chi(a) \not = 0 $ if $(a,n) = 1$.

Example 1: Let $p$ be an odd prime, define,
$$ \chi(a) = \left( \frac{a}{p} \right) \text{ by the Legendre symbol }$$
Then $\chi : \mathbb{Z} \to \mathbb{R} $ and it is a Dirichlet character mod $p$.

Example 2: Suppose that $n$ divides $m$ and $\chi$ is a character mod $n$. We can define a new character $\chi'$ mod $m$ by defining, $\chi'(a) = \chi(a)$ if $(a,m) = 1$ and $\chi'(a) = 0$ if $(a,m) = 0$. This character $\chi'$ is called the induced character of $\chi$.

A character mod $n$ for which there is no character mod $d$ ($d$ is a non-trivial divisor of $n$) induces $n$ is called a primitive character.

An application of these various quadratic symbols that you are learning about is that the only real-valued primitive characters are described by these symbols coming from quadratic reciprocity.

 

[Nono713]

All modern general purpose integer factorization algorithms work by finding small quadratic residues modulo a composite number, in order to construct a relation of the form $a^2 \equiv b^2 \pmod{n}$, producing a nontrivial factor of $n$ with good probability.