Regalo de otoño.

Por Carlos Pueyo Montañés

 







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Una solución sencilla.

Por Carlos Pueyo Montañés
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           El matemático francés Pierre de Fermat escribió un enunciado en 1.637 que decía algo así como...: “…No hay tres números enteros positivos a, b y c que cumplan la ecuación a ⁿ + b ⁿ = c ⁿ para ningún valor entero de n mayor de 2”.  No creo que sean las palabras exactas, pero sí: ésa es la idea.          Las excepciones son las soluciones triviales (0, 1, 1) , (1, 0, 1) y (0, 0, 0) , por supuesto.   Los casos n=1 y n=2 ofrecen infinitas soluciones: es algo que se sabe desde la antigüedad (dice Wikipedia).          Fermat no escribió la demostración de aquel enunciado porque, al parecer, no le quedaba bastante espacio en blanco en la página como para que le cupiera .            Tras el fallecimiento de Fermat, el libro donde escribió el enunciado debió de llegar a otras manos, ávidas por lee...
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A simple solution.

Por Carlos Pueyo Montañés
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                   French mathematician Pierre de Fermat wrote a statement around 1.637 A.D. that said “...no three positive integers a, b and c satisfy the equation a ⁿ + b ⁿ = c ⁿ for any integer value of n greater than 2” .          The exceptions are the trivial solutions (0, 1, 1) , (1, 0, 1) and (0, 0, 0) , of course.    The cases n=1 and n=2 have been known since antiquity to have infinitely many solutions (Wikipedia dixit).          Fermat didn’t write proof of his statement because, apparently, there was not enough blank space left in the page to do it .   Wait, what?          In 1995 (more than 350 years later) mathematician Andrew Wiles (with the help of his colleague Richard Taylor) proved Fermat’s statement in a published article…   The first complete, valid proof of that sta...
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Something I didn't think of.

Por Carlos Pueyo Montañés
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  I’m sorry.   There is something I didn’t think of some days ago, while I was looking for a simple solution to Fermat’s statement. I had just proved ( you can click here to see it ) that any triangle that has sides a, b, c the length of positive integers satisfies Fermat’s statement ( aⁿ ≠ bⁿ + cⁿ for any positive integer value of n > 2 ).   And I thought that was it… A mistake.   I mean, my proof was solid and sound; but I hadn’t finished yet.   Why? All triangles verify that each of its three sides is smaller than the sum (addition) of the other two. Let’s forget triangles from this point on. If we have three positive integers a, b, c …   One of them could be equal to the sum of the other two as in a = (b + c) . One of them could be bigger than the sum of the other two as in a > (b + c) . I need to prove that those two possibilities actually satisfy Fermat’s statement in order to have a complete, simple solution. And that’s what...
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