SOPHIE GERMAIN

Sophie Germain 

Matemática francesa nacida en 1776 que comenzó a interesarse por esta ciencia casi de casualidad. Según se cuenta, en la época de la Revolución Francesa se vivía un ambiente tan convulso que Sophie no podía salir de casa, por lo que leía libros de la biblioteca de su padre por puro entretenimiento. Gracias a uno de ellos conoció a Arquímedes, y su historia le llevó a seguir leyendo libros de matemáticas.

Sophie Germain fue una matemática autodidacta, y la forma que utilizó para difundir sus trabajos fue la correspondencia con otros matemáticos, algunos tan importantes como Joseph-Louis Lagrange y Carl Friedrich Gauss. Preocupada por el hecho de que pudieran no tomarla en serio por el hecho de ser mujer, en ambos casos lo hizo utilizando Monsieur LeBlanc como seudónimo. Tanto Lagrange como Gauss acabaron sabiendo que Monsieur LeBlanc era en realidad una mujer, pero a ninguno de ellos le importó lo más mínimo (en el buen sentido, se entiende).

Respecto a sus aportaciones a las matemáticas, Germain se dedicó principalmente a la teoría de números. Son importantes sus aportaciones sobre el último teorema de Fermat y sobre los números primos (de hecho, hay un tipo de números primos que se denomina primos de Germain). También es interesante destacar que, en geometría, introdujo el concepto de curvatura media de una superficie.

Una de las mayores contribuciones de Germain a la teoría de números fue la demostración matemática de la siguiente proposición: si x, y, z son enteros y x5 + y5 = z5, entonces al menos uno de ellos (x, y, o z) es divisible por cinco. Esta demostración, que fue descrita por primera vez en una carta a Gauss, tenía una importancia significativa ya que restringía de forma considerable las soluciones del último teorema de Fermat, el famoso enunciado que no pudo ser demostrado por completo hasta 1995.

Una de sus más famosas identidades, más comúnmente conocida como Identidad de Sophie Germain expresa para dos números x e y que:

x 4 + 4 y 4 = ( x 2 + 2 y 2 + 2 x y ) ( x 2 + 2 y 2 − 2 x y ) .   {\displaystyle x^{4}+4y^{4}=(x^{2}+2y^{2}+2xy)(x^{2}+2y^{2}-2xy).\ }

Intentó demostrar el Teorema de Fermat, y aunque no pudo hacerlo, obtuvo unos resultados que influyeron en las matemáticas de la época.

Así mismo, uno de sus resultados más conocidos es el conocido como Teorema de Sophie Germain, gracias a un pie de página en una obra de Adrien-Marie Legendre en 1823[6]​. Este teorema trata sobre la divisibilidad de las soluciones de la ecuación xp + yp = zp del Último teorema de Fermat para p primo impar. Sophie Germain probó que al menos uno de los números x, y, z tiene que ser divisible por p2 si puede encontrarse un primo auxiliar θ tal que se satisfacen las dos condiciones:

  1. No existen dos potencias p distintas de cero que difieran uno en modulo θ; y

  2. No existe ningún número tal que p sea potencia de orden p modulo θ de él.

En cambio, el primer caso del Último Teorema de Fermat (el caso en que p no divide xyz) tiene que cumplirse para cada primo p para el que pueda encontrarse un primo auxiliar. Germain identificó tal primo auxiliar θ para cada primo menor que 100

Abel Fernández

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