TITLE: How to explain to a layman why Fermat's Last Theorem involves non-trivial math?
QUESTION [8 upvotes]: Fermat's Last Theorem states, given$$x^n + y^n = z^n$$ no three integers $x,y,z$ will satisfy the equation given integer value of $n$ greater than two.
On the surface this seems like something that can be posed to a middle school student, whereby he will try a few cases for different values of $n$ and conclude that as $n$ goes large, there probably wouldn't exist a triple $x,y,z$ such that the equation will be satisfied.
What would be the first step of transforming this question to involve a non-trivial application of ring theory or group theory that Andrew Wiles' proof demonstrated? In other words, how do mathematicians see the connection between this equation and higher branches of mathematics? 
Are there similar examples i.e. estimating the next prime?

REPLY [2 votes]: Actually this is a good opportunity to explain to the layman how mathematics works, but I would change the motivating example a little bit.
Since we are searching for integer or rational solutions to the equation $x^n+y^n=z^n$, I would take a simple Diophantine equation and explain how to solve it using simple methods, for example parity (property of sums of integers), then change a little bit the equation in a way that our method fails, now we need a fancier method, then I would take yet another equation such that our last method fails too. At this point I would make a parallel explanation regarding why $x^n+y^n=z^n$ is a hard problem to solve.
PS$_1$: It's not hard to find problems in Diophantine equations such as the ones I talk above.
PS${_2}$: If your audience has a little bit of experience with mathematics you could actually explain why $x^n+y^n=z^n$ is hard using factorization techniques.