El título de esta entrada, por supuesto, se refiere al Último Teorema de Fermat para el caso n=4. Este es de hecho el caso más sencillo y del único que se conoce una demostración (implícita, o que se sigue como corolario) de Pierre de Fermat, y que además demuestra el Último Teorema para todo n divisible por 4. Uno pensaría que el caso n=3, por ejemplo, es uno más sencillo, sin embargo se pueden leer demostraciones más elaboradas en general, como aquí se verá para los casos de números primos impares (distintos de 2). La primera demostración del caso n=3 se atribuye a Leonhard Euler. Después se demostrarían varios casos más, hasta la aparición de Andrew Wiles, que acabaría de una vez por todas con la diversión. El documental de la BBC que aquí pongo es imperdible (también he dedicado antes una entrada a Yutaka Taniyama). Aunque aquí no seguiré precisamente sus pasos, Fermat demuestra implícitamente el caso n=4 de su Último Teorema al demostrar que el área de un triángulo rectángulo no puede ser un cuadrado. ¿Cómo es que se relaciona esto con el Último Teorema? Fermat demuestra que no existen soluciones de la ecuación ${x^2=z^4-y^4}$. Haré un bosquejo rápido para relacionar el área de un triángulo rectángulo con la ecuación anterior. Un triángulo rectángulo de catetos ${\alpha,\beta}$ e hipotenusa $\gamma$, por el teorema de Pitágoras, cumple ${\alpha^2+\beta^2=\gamma^2}$. Si además, pedimos soluciones enteras, la última es una ecuación diofántica, cuyas soluciones ${(\alpha,\beta,\gamma)}$ se llaman ternas pitagóricas. En esta entrada he mostrado que ${(2pq,p^2-q^2,p^2+q^2)}$ es terna pitagórica para ${p,q}$ coprimos y de paridad contraria. El área del triángulo rectángulo, como la conocemos desde la primaria, es $\displaystyle{\frac{\alpha\beta}{2}=pq(p^2-q^2)}$. Ahora bien, ${pq}$ y ${p^2-q^2}$ son coprimos, i.e. ${\mathrm{mcd}\left(pq,p^2-q^2\right)=1}$ (intenta demostrarlo por contradicción o de propiedades del mcd) y si el producto ${pq(p^2-q^2)}$ es igual a un cuadrado, en la entrada de ternas pitagóricas se ha mostrado que en estas condiciones, $p$ y $q$, (también ${pq}$ y ${p^2-q^2}$) son cuadrados. De aquí entonces, hagamos ${p=P^2}$, ${q=Q^2}$, de modo que para algún ${\zeta^2}$ se cumple ${\zeta^2=pq(P^4-Q^4)}$ y como $pq$ también es un cuadrado, digamos ${pq=\xi^2}$, se sigue que $\displaystyle{\left(\frac{\zeta}{\xi}\right)^2=P^4-Q^4}$. De ese modo Fermat razona que si el área de un triángulo rectángulo fuera un cuadrado, entonces existiría un par de potencias cuartas cuya diferencia sería un cuadrado. Al probar Fermat que el área de un triángulo rectángulo no puede ser un cuadrado, estaría además probando el Último Teorema para n=4, ya que si ${x^2=z^4-y^4}$ no tiene soluciones enteras, entonces ${\left(x^2\right)^2+y^4=z^4}$ tampoco tiene soluciones enteras. Así pues, hay que demostrar que ${x^4+y^4=z^2}$ no tiene soluciones enteras, y uno de los procedimientos más sencillos es precisamente siguiendo un camino parecido al que se siguió para encontrar las ternas pitagóricas. En general serán útiles algunas nociones sencillas de la entrada de ternas pitagóricas que aquí daré por sentado. Se sabe que si la ecuación anterior tiene soluciones, éstas son ternas pitagóricas, $${(x^2,y^2,z)=(2pq,p^2-q^2,p^2+q^2)}$$ suponiendo sin pérdida de generalidad que ${x^2}$ es el término par. Bien, se sigue entonces que ${p^2=y^2+q^2}$, por tanto ${(q,y,p)}$ es otra terna pitagórica; asumamos ahora sin pérdida de generalidad que $q$ es el término par, de modo que para ${P,Q}$ coprimos y de paridad contraria, se tiene la terna pitagórica ${\left(2PQ,P^2-Q^2,P^2+Q^2\right)}$. Así pues, en términos de esta última terna, se tiene $$x^2=4PQ\left(P^2+Q^2\right)\hspace{0.25in}\text{i.e.}\hspace{0.25in}{PQ\left(P^2+Q^2\right)=\left(\frac{x}{2}\right)^2}$$ Ahora bien, ${\mathrm{mcd}(P,Q)=\mathrm{mcd}\left(PQ,P^2+Q^2\right)=1}$ (propiedades del mcd), y también, sabemos que ${PQ}$ y ${P^2+Q^2}$ por ser coprimos y cuyo producto es un cuadrado, ellos mismos son cuadrados, y así también $P$ y $Q$ son cuadrados, digamos que ${\rho^2=P}$, ${\delta^2=Q}$ y $\vartheta^2={P^2+Q^2}$ de modo que $$\rho^4+\delta^4=P^2+Q^2=\vartheta^2=p < p^2+q^2=z < z^2$$ es decir $$\vartheta^2 < z^2$$ esto es $$x^4+y^4=z^2\;\Longrightarrow\;\rho^4+\delta^4=\vartheta^2,\;\vartheta < z$$ y así también que exista la terna ${(\rho^2,\delta^2,\vartheta)}$ implica que existe otra ecuación con las mismas características con algún ${\epsilon<\vartheta}$, lo que implica que existe otra ecua... y así ad infinito, lo cual es imposible, ya que sabemos que los números naturales restantes son finitos o mejor dicho, los números naturales en general son bien ordenados, por lo que eventualmente no se encontrará otra ecuación que satisfaga lo anterior. A este tipo de demostración se le llama por descenso infinito, y de hecho se atribuye a Pierre de Fermat. Como se ha dicho, con esta demostración también de demuestra que ${x^4+y^4=z^4}$ no tiene soluciones enteras. Se asume que de aquí surgiría el Último Teorema de Fermat, cuyo nombre alude a que sería la única aseveración de Fermat que carecía de demostración, i.e. ya todos los teoremas de Fermat estaban demostrados, sólo quedaba un último teorema por demostrar. Claramente se puede extender aún más este resultado. Primero, el Último Teorema de Fermat es cierto para n divisible por 4, es decir, para algún $k$ tal que ${n=4k}$, de modo que $$\left(x^{k}\right)^4+\left(y^k\right)^4=\left(z^k\right)^4$$ en particular, ${2^u|4k}$ siempre que ${u>1}$, por lo que se confirma que el teorema de Fermat es cierto para potencias de dos. Esto es relevante, pues si existen soluciones de ${x^n+y^n=z^n}$, entonces n no puede ser potencia de dos, lo que implica que existe un primo ${p\neq{2}}$ tal que ${p|n}$, es decir, existe un $\kappa$ tal que ${n=p\kappa}$ y se buscaría resolver $$\left(x^\kappa\right)^p+\left(y^\kappa\right)^p=\left(z^\kappa\right)^p$$ y así, con el trabajo realizado, el Último Teorema de Fermat estaría demostrado (o confirmado), si se pudiera demostrar para cada primo ${p\neq{2}}$. Los primeros pasos en esta dirección fueron dados por Euler para ${p=3}$, Dirichlet y Legendre para ${p=5}$, etc... lo demás es historia (una gran historia).
Showing posts with label Último teorema de Fermat. Show all posts
Showing posts with label Último teorema de Fermat. Show all posts
Yutaka Taniyama
The following is an excerpt of the article by J J O'Connor and E F Robertson.
Read the original @ www-history.mcs.st-andrews.ac.uk/Biographies/Taniyama
YUTAKA TANIYAMA
Born: 12 Nov 1927 in Kisai (north of Tokyo), Japan
Died: 17 Nov 1958 in Tokyo, Japan
His parents were Sahei, a medical doctor, and Kaku Taniyama. Yutaka was born into a large family having two older brothers and three older sisters as well as a younger brother and a younger sister. Yutaka was a sickly child and suffered from tuberculosis which caused him to miss two years of high school. After graduating from the high school, he entered the University of Tokyo to study mathematics. During his undergraduate years he read Claude Chevalley's Theory of Lie groups and André Weil's Foundations of algebraic geometry as well as two other books by Weil on algebraic curves and abelian varieties. He attended algebra lectures by Masao Sugawara and these encouraged his interest in number theory. He graduated in March 1953.
He remained at the University of Tokyo as a 'special research student' in the Department of Mathematics, although he had no thesis advisor. Shimura writes in [1] about the apartment where Taniyama lived in Tokyo:
Taniyama's interests were in algebraic number theory and his fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo and Nikko in 1955. His meeting with André Weil at this symposium was to have a major influence on Taniyama's work. These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field. This conjecture proved to be a major factor in the proof of Fermat's Last Theorem by Andrew Wiles. In the Proceeding of the conference he published the paper Jacobian varieties and number fields, then in the following year the paper L-functions of number fields and zeta functions of abelian varieties.
Other than these two papers the only other paper Taniyama published was Distribution of positive 0-cycles in absolute classes of an algebraic variety with finite constant field (1958). However, in addition to these papers, he wrote the book Modern number theory (1957) in Japanese, jointly with G Shimura.
With seemingly a great future in front of him, both in mathematics and his life (he was planning marriage to Misako Suzuki) he took his own life. In a long suicide note he left, he took great care to describe exactly where he had reached in the calculus and linear algebra courses he was teaching and to apologise to his colleagues for the trouble his death would cause them. As to the reason for taking his life he says:
About a month later his fiancé Misako Suzuki also committed suicide. She left a note which included the sentences:
Shimura writes [1]:
One might reasonably ask what Taniyama's interests were other than mathematics. He enjoyed listening to music, especially Beethoven's Eighth Symphony, and going to movies, his favourite film being 'The King and I'. His only hobby was writing articles which he never intended to publish, but he seemed to find writing them helped to organise his thoughts. Examples of the topics he wrote articles on included: reviews of books, ideas on how researchers should be trained, how to organise a new institute for mathematical sciences, and reviews of articles by others.
Full article by: J J O'Connor and E F Robertson
April 2009
Read the original @ www-history.mcs.st-andrews.ac.uk/Biographies/Taniyama
YUTAKA TANIYAMA
Born: 12 Nov 1927 in Kisai (north of Tokyo), Japan
Died: 17 Nov 1958 in Tokyo, Japan
His parents were Sahei, a medical doctor, and Kaku Taniyama. Yutaka was born into a large family having two older brothers and three older sisters as well as a younger brother and a younger sister. Yutaka was a sickly child and suffered from tuberculosis which caused him to miss two years of high school. After graduating from the high school, he entered the University of Tokyo to study mathematics. During his undergraduate years he read Claude Chevalley's Theory of Lie groups and André Weil's Foundations of algebraic geometry as well as two other books by Weil on algebraic curves and abelian varieties. He attended algebra lectures by Masao Sugawara and these encouraged his interest in number theory. He graduated in March 1953.He remained at the University of Tokyo as a 'special research student' in the Department of Mathematics, although he had no thesis advisor. Shimura writes in [1] about the apartment where Taniyama lived in Tokyo:
... he lived in a one-room apartment which consisted of 81 square feet of living space, a sink, and a tiny unfloored part behind the door. Running water, gas and electricity were provided separately in each room, but there was only one toilet on each floor of the two-storey building, shared by all the occupants of the dozen or so rooms of the floor. I remember that his was No. 20 on the second floor, close to the last. Thus it was more like a dormitory than an apartment, but it was more or less typical of the time. To take a bath, he had to go to a public bathhouse, a few minutes' walk from his apartment. The building, a rather shabby wooden structure, was named poetically 'Villa Tranquil Mountains'...
Taniyama's interests were in algebraic number theory and his fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo and Nikko in 1955. His meeting with André Weil at this symposium was to have a major influence on Taniyama's work. These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field. This conjecture proved to be a major factor in the proof of Fermat's Last Theorem by Andrew Wiles. In the Proceeding of the conference he published the paper Jacobian varieties and number fields, then in the following year the paper L-functions of number fields and zeta functions of abelian varieties.
Other than these two papers the only other paper Taniyama published was Distribution of positive 0-cycles in absolute classes of an algebraic variety with finite constant field (1958). However, in addition to these papers, he wrote the book Modern number theory (1957) in Japanese, jointly with G Shimura.
With seemingly a great future in front of him, both in mathematics and his life (he was planning marriage to Misako Suzuki) he took his own life. In a long suicide note he left, he took great care to describe exactly where he had reached in the calculus and linear algebra courses he was teaching and to apologise to his colleagues for the trouble his death would cause them. As to the reason for taking his life he says:
Until yesterday I have had no definite intention of killing myself. But more than a few must have noticed I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be some to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing all my life.
About a month later his fiancé Misako Suzuki also committed suicide. She left a note which included the sentences:
We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him.
Shimura writes [1]:
...he was the moral support of many of those who came into mathematical contact with him, including of course myself. Probably he was never conscious of this role he was playing. But I feel his noble generosity in this respect even more strongly now than when he was alive. And yet nobody was able to give him any support when he desperately needed it. Reflecting on this, I am overwhelmed by the bitterest grief.
One might reasonably ask what Taniyama's interests were other than mathematics. He enjoyed listening to music, especially Beethoven's Eighth Symphony, and going to movies, his favourite film being 'The King and I'. His only hobby was writing articles which he never intended to publish, but he seemed to find writing them helped to organise his thoughts. Examples of the topics he wrote articles on included: reviews of books, ideas on how researchers should be trained, how to organise a new institute for mathematical sciences, and reviews of articles by others.
Full article by: J J O'Connor and E F Robertson
April 2009
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