Here it is:

If $x, y, z \text{ and } n$ are all natural numbers with $n>1$ and $x^n + y^n = z^n$, prove that $x, y, z > n$.

So, how do we go about this? $x^n + y^n = z^n$ seems like a pretty difficult equation to mess about with (Fermat's Last Theorem, anyone?), but, we might gain some insight if we move some stuff around, plug in a few numbers, etc.

But, we should try to find out some more information by just observing the problem; it might be useful.

Obviously, $z > x,y$ (from the equation given). And, we can also say, without losing generality, $y \ge x$, since the equation is symmetric about these two terms (i.e. we can interchange $x$ and $y$, and it won't make any difference whatsoever).

Let's try to separate out one of the variables, and see what we can do.

$x^n = z^n-y^n$

If you remember some factorization tools:

$x^n = (z-y)(z^{n-1} + z^{n-2}\cdot y + z^{n-3} \cdot y^2 + ... + z^{1}\cdot y^{n-2} + y ^ {n-1})$

Since $z > y$ is the same as $z > y+1$ since $z, y$ are natural numbers:

$x^n > ((y+1)-y)(z^{n-1} + z^{n-2}\cdot y + z^{n-3} \cdot y^2 + ... + z^{1}\cdot y^{n-2} + y ^ {n-1})$


$x^n > ((y+1)-y)(nx^{n-1}) = nx^{n-1}$

So, finally,

$x > n$

And, since the equation is symmetric for $x$ and $y$, we get $y > n$, and we can use very similar methods to prove $z > n$.

So, what are some important moves here?

First of all, noticing the symmetric nature of the equation helped us get down some initial observations and assumptions, which made our lives much easier.

Factoring the $z^n - y^n$ allowed us to tackle it much more effectively.

Then, going from an equation to an inequality by replacing $z$ with $y+1$ and simplifying the expansion to having terms of $x$ allowed us to complete the proof.

But, the biggest move was just separating out the $x$ on the left side so that we would be able to use methods on the right side.

This problem would probably be considered as part of number theory, which has a host of very interesting and easy to understand problems that don't require much knowledge to solve, but, require quite a measure of creativity.

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