## Fermat's Last Theorem (*) ### Introduction

Fermat's Last Theorem states the following:

«$$2$$ is the only value of $$n$$ (a natural number bigger than one), for which the equation $$x^{n}+y^{n}=z^{n}$$ has solutions in which $$x, y, z$$ are (all of them) positive integers».

For $$n=2$$, the triple $$x=3, y=4, z=5$$ is a solution; another one is $$x=8, y=15, z=17$$. But, for instance, for $$n=4$$, one can say that, for any natural numbers $$x, y, z$$, we have $$x^{4}+y^{4}\neq z^{4}$$.

Taking, for each natural $$n$$, the surface defined by the points in space whose coordinates satisfy the identity $$x^{n}+y^{n}=z^{n}$$, one may give a geometrical interpretation of the statement of Fermat's Last Theorem. The applet in this page illustrates this interpretation. For more information about the Theorem, consult the page "Fermat's Theorem".

A natural question associated with Fermat result is about the number of (integer) solutions corresponding to a given $$z$$, when $$n=2$$, or, more generally, about the number of integer solutions of the equation (in $$x$$, $$y$$) $$x^{2}+y^{2}=m$$, for each natural number $$m$$. These questions are related with several other interesting problems, some of them solved by Gauss himself. Meanwhile, you may play with the applet below.