Saturday, August 27, 2005

Sophie Germain

Sophie Germain was born on April 1, 1776 at the dawn of the French Revolution. Her father was a very successful merchant who was able to avoid the great unrest of the revolution.

When Sophie was 13, she discovered a book on the history of mathematics in her father's library. She became fascinated with the story of Archimedes who was killed by a Roman soldier while he was working on a geometry problem in the sand. Later, she became interested in the works of Leonhard Euler and Sir Isaac Newton. Her parents thought it was inappropriate for a young girl to be so interested in mathematics and tried to discourage her. There is a story where they tried to hide her candles and her warm clothes so that she would not be able to study math at night.

In 1794, when Sophie was 18, the Ecole Polytechnique opened in Paris. This was a school founded upon the principle of training France's most talented mathematicians and scientists. Unfortunately for Sophie, no women were allowed to enroll. Sophie found a way around this restriction. She learned of a Monsieur LeBlanc who had enrolled in the Polytechnique but was now traveling out of Paris. She took on his identity, picking up his assignments, and handing in his homework. The plan went very well until some of her math work caught the attention of the mathematician Joseph Louis Lagrange.

Lagrange was one of the most famous mathematicians of his day. He had worked closely with M. LeBlanc and was now surprised to see a student which had showed little aptitude for mathematics previously was now doing outstanding work. He requested to set up a meeting. It was at this point that Sophie admitted her ruse. Lagrange was quite impressed.

It was perhaps this experience with Lagrange that led Sophie to start a correspondence with the most famous mathematician of the day, Carl Friedrich Gauss in 1804. Uncertain if Gauss would respond well to a woman, she again used the identity of M. LeBlanc. Gauss was quite impressed with the content of the letter and from this point on, Gauss and Sophie continued to exchanged letters on a regular basis.

It was because of Napoleon in 1807 that Gauss found out about Sophie's true identity. Napoleon's army was heading into the Brunswick where Gauss lived. The invading force was led by General Pernety who was a personal friend of Sophie's. She asked the General to ensure Gauss's safety. The general told Gauss that he was under the protection of one Sophie Germain who Gauss had never heard of. After this, Sophie admitted her true identity.

The response by Gauss is often quoted (the translation below is taken from Wikipedia):

But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and superior genius. Indeed nothing could prove to me in so flattering and less equivocal manner that the attractions of this science, which has enriched my life with so many joys, are not chimerical, the predilection with which you have honored it.
Sophie became fascinated in particular with Fermat's Last Theorem. She developed a new line of attack. Rather than proving that there were no solutions to a given value n, she showed that if there was a solution, a certain condition would have to apply. This insight would later lead to the proof for Fermat's Last Theorem where n=5.

In 1811, Sophie submitted an entry to a contest sponsored by the French Academy of Sciences. The goal was to provide a solution to a fundamental mathematical problem in the physics of elasticity. Sophie's was the only submission for that year. Unfortunately, her solution was not accepted. In 1816, on her third submission, Sophie's solution was accepted. This was a very important achievement in establishing Sophie Germain as a professional mathematician. Her paper would later become one of the foundations of the modern theory of elasticity.

Throughout her life Sophie Germain never married. She relied on the support of her father in order to continue her studies in mathematics. She became the first woman invited to attend the Academy of Sciences sessions. In 1830, she was awarded an honorary PhD.

Sophie Germain died in 1831 at the age of 55 after a two year struggle with breast cancer. Today, Sophie Germain is considered one of the most talented mathematicians of all time.


Wednesday, August 24, 2005

Another False Proof: Russian Professor claims to have solved Fermat's Last Theorem

Today, I learned of a news story that a professor in Russia claims to have solved Fermat's Last Theorem in 3 lines. The story has also been covered in Italy here and here. (For those who don't speak Russian, here is an English translation. For those who don't speak Italian, here and here are English translations of the Italian articles).

It looks like the Russian article is the only one that gives the details of the proof. So, I am relying on this translation.

Let me start by saying that it is very important to be skeptical of anyone claiming to have a solution. In May, the Manilla Times stated that a local man found a mistake in the proof by Andrew Wiles. This of course turned out to be a hoax.

There are two tell-tales sign that the proof is most likely flawed. First, the professor who claims to have found the proof is not an expert in number theory. Second, the proof has not been reviewed by anyone who is an expert of number theory. From the mathematician Gabriel Lame to the very famous Leonhard Euler, it is very easy to make mistakes when it comes to number theory. Even Fermat made a mistake when it came to his conjecture about what are today known as Fermat primes.

Now, onto his proposed proof. After reviewing an English translation of the article, I found (no surprise) that the proof is not a valid demonstration of Fermat's Last Theorem.

First, let me summarize the proof:

(1) Assume that Fermat's Last Theorem is true.
(2) Then, there exists xn + yn = zn with n ≥ 3.
(3) For any x,y, we can create a right triangle and so we can suppose a value r such that:
r2 = x2 + y2.
(4) Since n ≥ 3, we know that z is less than r. [I will add the details for this later]
(5) We can construct a triangle based on z,x,y. Since z is less than r, we know that the angle opposite z (let's call it angle B) must be less than 90 degrees (and of course, greater than 0 degrees).
(6) Now, using the Law of Cosines, we know that:
z2 = x2 + y2 - 2xycosB.
(11) Since B is greater than 0 and less than 90, we know that cosB cannot be a whole number. [See here for a Cosine look up table]


OK, so if the steps are valid, is Fermat's Last Theorem proven? Not at all. Just because cos B is not a whole number doesn't prove that z is not a whole number. That is ultimately what must be proven.

The professor must either prove that cos B is irrational or he must show that based on 2xycosB, z cannot be a whole number.

Since cos B is a continuous function, it necessarily covers rational values and therefore it is quite possible that 2xycosB is a whole number.

Thanks go out to David (see the comments section) for providing his thoughts.