TITLE: What is the point of quadratic residues?
QUESTION [10 upvotes]: 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?

REPLY [5 votes]: I always like to motivate them through the want to solve Diophantine equations.
If I challenge someone to solve any of these nasty equations over the integers:
$x^2 + 440xy^{45} + 88x^{57} = 3$
$48x^5 + y^2 + 3000z^{34} = 17$
$55x^4 + 60xy^3 + 50001z^2 = 1532877192878$
then you will surely struggle by using elementary means. In each case you could set up a brute force search and get no solutions but of course this isn't really a proof.
However you notice that if there were any solutions to these then you could reduce each equation mod $4,3,5$ respectively and would be left solving:
$x^2 = 3 \bmod 4$
$y^2 = 2 \bmod 3$
$z^2 = 3 \bmod 5$
and these really ARE things we can show have no solutions very easily! It only takes a finite amount of searching to find quadratic residues. The great thing here is that I could have changed most of the coefficients above by multiples of $4,3,5$ and still had no solutions.
The power of modular arithmetic is amazing.
Another nice little result is that no number that is $3 \bmod 4$ can ever be a sum of two square numbers. Why is this? Well if $n \equiv 3 \bmod 4$ and $x^2 + y^2 = n$ then looking mod $4$ the LHS would have to be $0,1$ or $2$ (since the quadratic residues mod $4$ are $0,1$) yet the RHS would be $3 \bmod 4$!