ATRACTOR

Pierre de Fermat (1601-1665), a French lawyer at the Parliament of Toulouse, was a mathematician known in particular by his works in number theory. The famous "Fermat's Last Theorem" states that there is no solution for the equation \(x^{n}+y^{n}=z^{n}\), if \(n\) is an integer greater than \(2\) and \((x, y,z)\) are integers \(> 0\).

Fermat wrote that statement in the margins of his copy of the "Arithmetica" of Diophantus and marked that he had found "a truly marvelous proof of this proposition which this margin is too narrow to contain." This conjecture remained unproved for a long time and constituted a challenge for mathematicians over the ages. Along these centuries, numerous people announced the proof of Fermat's conjecture, but errors have been found, in most cases quite coarse.

Finally, in June 1993, Andrew Wiles, a british mathematician from Princeton University (USA), presented at a seminar in Cambridge what he believes to be a proof of Fermat's Last Theorem, a result of his work of 7 years on the conjecture. The printed version of his proof had some 200 pages, but a failure in it was found later on. In collaboration with his former student Richard Taylor (Cambridge Univ., UK), Wiles was able to circumvent the problem and completed the proof in a second paper. Both papers were published in May 1995 in a dedicated volume of the Annals of Mathematics. This was the conclusion of the proof of Fermat's Last Theorem, some 350 years after Fermat stated it. It is somewhat curious that the result, even before being proved, has always been known as Fermat's Last Theorem and not Fermat's Last Conjecture, as it would be more accurate.

The applet provides a geometric interpretation of the statement. It shows a surface that represents the set of points \((x,y,z)\) such that \(x^{n}+y^{n}=z^{n}\), with natural \(n\) *, and an horizontal plane (formed by points \((x,y,0)\)), in which the lines that correspond to integer values for \(x\) and for \(y\) are drawn in white. The vertices of the white squares are precisely the points in the plane with integer coordinates. In the surface it is drawn in yellow one of the level curves **, of integer height \(z\). The level curve is modifiable in the applet - see Help-Fermat . The applet also shows the projection of the level curve in the horizontal plane. The surface points with integer coordinates are also marked, as well as the vertical lines that link them to their projections in the horizontal plane. Initially \(n\) is equal to \(2\) and the applet has the following look:

The points marked in the surface represent the integer solutions (only the positive ones in case the option is selected) of the equation \(x^{n}+y^{n}=z^{n}\), with natural \(n\). Notice that the referred points are precisely the ones that belong to a level curve of integer height and project into a vertex of some white square in the plane. In other words, each integer solution of the equation corresponds to one vertex of a white square that intersects one of the yellow curves (the projection of a level curve with integer height).

Note: In the figures marked with , if you put the mouse over it you will see some animated gif.

For \(n = 4\) and \(Limit=3.0\)

Both points and lines may have gray or black colour. The gray points are the solutions in which at least one of the coordinates is zero. For every natural \(n\), the equation \(x^{n}+y^{n}=z^{n}\) has an infinite number of solutions of that kind, for example \((a,0,a)\), with \(a\) a natural number. Therefore, for each natural \(n\), one finds always points and the corresponding lines in gray. However, such fact does not contradict the statement of the Theorem of Fermat, since the coordinates of these points, although being integer solutions of \(x^{n}+y^{n}=z^{n}\), are not all positive numbers. Try to select the option in the menu : the surface will be represented by positive values, and notice that then there will be no gray points and lines.

For \(n = 3\) and \(Limit = 4\).

The points of black colour represent the integer solutions in which all coordinates are nonzero, that is, a black point belongs to some level curve of integer height and projects into the vertex of a white square that is neither in the plane \(x=0\) nor in the plane \(y=0\). Accordingly to Fermat's Theorem, there are only black points when \(n =2\).

Level curve of height \(5\), for \(n = 2\).