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.

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