[sxhydedb] Why are integers special?

Fermat's Last Theorem seemed silly and trivial to me when I first encountered it as a child: it was very obviously false with plenty of counterexamples with real numbers.  Later, I learned of the restriction to integers, but it still seemed a little bit silly: the restriction seemed artificial and arbitrary just to make the problem difficult.

Why restrict yourself to integers, especially for an equation like Fermat's which can easily accept reals?  One could argue that the integers are merely points on the real number line that happen to have compact names, but there isn't anything more special about them, assuming the primacy of the reals.

I want to travel the world but restrict myself only to national capitals.