What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues?
Why is the Law of Quadratic Reciprocity considered as one of the most important in number theory?
[Math] the point of quadratic residues
educationelementary-number-theorymotivationquadratic-reciprocityquadratic-residues
Best Answer
Warning: these are pure math examples of why we like quadratic residues, not real life.
Well, this is more quadratic residues than quadratic reciprocity, but the computation of $\left(\frac{-1}p\right)$ and $\left(\frac{-3}{p}\right)$ (those are Legendre symbols) are essential to determining when primes in the natural numbers are prime in the Gaussian integers ($\Bbb Z[i]$) and the Eisenstein Integers ($\Bbb Z[e^{2\pi i/3}]$). They are furthermore necessary for the proof of when numbers are expressible as $a^2+b^2$ or $a^2+ab+b^2$ for $a,b\in\Bbb Z$. The two-square proof is also extendible to Lagrange's Four Square Theorem, which also uses quadratic residues.
There is also a pretty cool relationship between Jacobi symbols and permutations which is that if you write the action of multiplication on a ring $\Bbb Z/n\Bbb Z$ by $a$ in cycle notation (e.g. $2\cdot \Bbb Z/9\Bbb Z$: $$ \{0,1,2,3,4,5,6,7,8\}\to\{0,2,4,6,8,1,3,5,7\}=(0)(124875)(36) $$) then the sign of the permutation (where sign is defined to be $(-1)^{n}$ where $n$ is the number of $2$-cycles the permutation decomposes into) is equal to $\left(\frac an\right)$, where that is the Jacobi symbol.
Stated more concisely: $\text{sgn}(a\cdot(\Bbb Z/n\Bbb Z))=\left(\frac an\right)$ where $a\cdot(\Bbb Z/n\Bbb Z)\in S_n$.
Lastly, I'll refer you to this mathSE post which has a lot of good answers to a similar question.