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.

For more information click here.

This applet uses Javaview.

Translated for Atractor by a CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

(*) This work was carried out under a grant by FCT - Fundação para a Ciência e a Tecnologia.

Difficulty level: University