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.

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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.