En mi primera semana sabática de 2011 estuve trabajando en algunos temas matemáticos (ver Cinco al hilo). En realidad demostré cuatro proposiciones, y fallé en encontrar una demostración para el clásico último teorema de Fermat, para n = 3. Una de los teoremas que demostré (todos los primos de la forma 4n+1 se pueden expresar como suma de dos cuatrados) me llevó unos días, en el último paso, donde creo recordar que apliqué el descenso infinito (ver Proof by infinite descent, Fermat's Infinite Descent) . La demostración de Gauss de Fermat n=3 también emplea el mismo método, y aún hoy no encontré el camino para ese último paso que me falta.
En estos días me encuentro con un texto de Fermat, donde comenta su método:
"Since the ordinary methods which are in books were insufficient to prove such difficult propositions, I finally discovered an altogether singular method to succeed. I called this method of proof indefinite or infinite descent: at first I used it only to prove negative propositions, such as for example:
That there is no number, less by a unit than a multiple of 3, which is composed of a square and the triple of another square;
That there is no right triangle in numbers whose area is a square number. The proof is done by reduction to the absurd in this manner:
If there were a right triangle with integral sides whose area was equal to a square, there would be another triangle smaller than this one which would have the same property. If there was a second, smaller than the first, with the same property, then by the same reasoning, there would be a third, smaller than the second, which would have the same property, and then a fourth, a fifth, and so on descending to infinity. However, given a number, there is no way to descend from it infinitely (here I only speak of integers). Hence we conclude that it is impossible to have a right triangle whose area is a square.
We infer from this that neither is it possible to have a triangle whose sides have fractional (rather than integral) lengths and whose area is a square. Because if we had such a triangle with fractional sides, then we could construct one with integral sides, which is in contradiction with what we proved above.
Vean que Fermat no explica en detalle el paso de descenso infinito:
I do not add the reason for which I infer that if there were a right triangle of this nature, there would be another of the same nature smaller than the first, because the discourse would be too long and therein lies the whole mystery of my method. I would be greatly satisfied if Pascal and Roberval and many other scholars sought it upon my indications.
Pero agrega otro ejemplo, relacionado con uno de los teoremas que mencioné antes:
I remained for a long time unable to apply my method to positive assertions, because the proper angle to approach such questions is much more uncomfortable than the one which I use for negative assertions. Thus, when I needed to show that every prime number which surpasses a multiple of 4 by a unit is composed of two squares, I found myself in a quandary. But finally, much-renewed meditation provided the light which I lacked, and positive assertions became solvable by my method, with the help of some new principles which necessity forced me to add to it. This progress in my reasoning on such positive assertions runs as follows: if a prime number chosen at will, which surpasses a multiple of 4 by a unit, is not composed of two squares, there will be a prime number of the same nature, less than the given one, and then a third still less, etc., descending to infinity until one arrives at the number 5, which is the least of all primes of this nature, which shows that it cannot be composed of two squares, which yet it is. Hence we infer, by deducing the impossible, that all primes of this nature are, consequently, composed of two squares".
Tendría que escribir un post sobre la prueba por descenso infinito de la irracionalidad de la raíz cuadrada de 3. Así como de todos los primos 4n+1 como suma de dos cuadrados. Podría agregar alguna demostración usando ese método de la no existencia de triángulos rectángulos con lados enteros y superficie cuadrado perfecto.
El texto de arriba lo encontré en el mismo libro que me sirvió en el post de ayer sobre Pascal y el avance de la ciencia: el "Invitation to the Mathematics of Fermat-Wiles", de Yves Hellegouarch.
Angel "Java" Lopez