Tuesday, December 27, 2011

44: Friends In Mathematical Places

    Before we start, I'd like to thank listener Sean Monahan for
suggesting today's topic.  By the way, everyone listening out there,
don't forget to put yourself on the frapper map at mathmutation.com,
and email me at erik @ mathmutation.com to say hi.  Maybe you too will
achieve fame and fortune as I mention your name in the podcast.
    Anyway, now on to today's topic.  When reading the classic
"Fermat's Enigma" by Simon Singh, you might be surprised to find out
that as far back as the early 19th century, major contributions to
solving Fermat's Last Theorem were made by a female mathematician,
Sophie Germain.  Sophie Germain was the daughter of a middle-class
merchant family in France.  After reading a book about Archimedes at
the age of 13, while locked up in her home hiding from the French
Revolution, she decided to teach herself mathematics, despite the
opposition of her parents.  Back then it was unthinkable for a woman
to enter that field.  After studying lecture notes from classes she
could not attend and posing as a man in correspondence for many years,
she won a major competition at the French Academy of Sciences.  After
proving some important results related to Fermat's Last Theorem, she
became the first woman to attend sessions there, not counting wives
accompanying their husbands.   
    Her major contributions included a proof that if x, y, and z are
integers, then for any solution of x^5 + y^5 = z^5 must have the
property that x, y, or z is divisible by 5.   As you will probably
recognize, this is a special case of the equation described by
Fermat's Last Theorem, x^n + y^n = z^n, and her result led to a proof
of that theorem for the special case of n=5.  More generally, this led
her to the result that if n is an odd prime and 2n+1 is prime, there
could be no solution to the equation of Fermat's Last Theorem unless
x, y, or z is divisible by n.  Her theorem was actually a little more
general than this, and you can find a detailed description referenced
in the show notes.
    The primes mentioned in her result, prime numbers with the
property that if you double them and add 1, you get another prime, are
now known as "Sophie Germain Primes".  It is still an open question
whether there are an infinite number of these special primes, or if
there is some largest one after which no more exist.  If there is a
largest one, though, it is pretty big-- currently one is known that
has 51,910 digits.
    Probably the most amusing story from Germain's biography is how
she saved the life of legendary German mathematician Carl Freidrich
Gauss.  She had been corresponding with him for several years on
mathematical topics using her male pseudonym, Monsieur Le Blanc,
though they had never met.  In 1806, when Napolean invaded Prussia,
she was afraid that not realizing his importance, soldiers would kill
him during the invasion, the same way her hero Archimedes had
perished.  So she pulled some strings and had a French general of her
acquaintance personally guarantee Gauss's safety.  Finding a protector
in the French army, Gauss was quite surprised, especially since the
general mentioned he was doing Sophie Germain a favor, and he had
never heard of this woman.  Finally she revealed her true identity to
Gauss, and he was astonished to find his respected colleague LeBlanc
was actually female. 
    Gauss eventually convinced the University of Gottingen to award
Germain an honorary degree, in place of the true degree she should
have received but could not at that time due to her gender.  Sadly she
died of breast cancer before she could accept it.
    And this has been your math mutation for today.

  • Sophie Germain at Wikipedia
  • Sophie Germain's Contributions to Fermat's Last Theorem
  • Fermat's Enigma at Wikipedia
  • Germain biography at About.Com
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