Wednesday, January 25, 2006

Lamé's Proposed Proof

On March 1, 1847, Gabriel Lamé announced that he believed that he had found a full proof for Fermat's Last Theorem. He presented to the Paris Academy the outline of what he believed was a complete proof. Earlier he had succeeded in the first proof for x7 + y7 = z7.

The approach that he offered involved what are known as the Roots of Unity. This involves complex numbers such as α where:

αn = 1 and α ≠ ± 1.

The proof is based on the following equation:

xn + yn = (x + y)(x + αy)(x + α2y) .... (x + αn-1y)

The essence of the proof is showing that each of the x+αey is relatively prime to the others so that it represents an nth power (since they are all equal to zn). Then, one shows how this condition leads to infinite descent.

Lamé claimed that this work was largely based on the work done by Joseph Liouville.

Liouville was present at the Paris Academy at this time and spoke after Lamé. He was not as confident as Lamé about the success of the proof. In fact, he suggested that Lamé approach was rather obvious and would have occurred to anyone who had worked long enough with complex numbers.

Liouville claimed that Lamé's assumption about unique factorization of the roots of unity was not necessarily justified and absent unique factorization, the proof was not valid.

The next person to speak was Augustin Cauchy. Cauchy was more optimistic about the possibilities of Lamé's approach. He announced that he too was very close to a solution to Fermat's Last Theorem using a similar technique.

On March 15, Pierre Wantzel announced that he had a proof about the unique factorization of the roots of unity. His proof showed unique factorization for the case where n = 2 and n=3. He then claimed that "one easily sees" the validity for cases where n ≥ 4. In a short time, Cauchy came back demonstrating that Wantzel's method did not work for n ≥ 4.

At this point, a competition broke out between Lamé and Cauchy to prove unique factorization for n ≥ 4. On March 22, both men deposited "secret packets" to the Paris Academy, a method that was used to establish priority.

Unfortunately, as is usually the case, the devil is in the details and the proof for unique factorization turned out to be more complicated than Lamé and Cauchy had forseen. In May 24, Liouville announced a proof by Ernst Kummer that settled the question by demonstrating that Lamé's complex numbers lacked unique factorization. In fact, Kummer had first presented this proof in a mathematical journal three years earlier.

But there was hope. Kummer's conclusion was that unique factorization could be "saved" by using "ideal complex numbers." Kummer's ideal complex numbers would turn out to be a major breakthrough in the generalization of Fermat's Last Theorem. It would also turn out to be the foundation for what is today known as algebraic number theory.

Today's blog was taken from Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. (All quotes come from Edwards' original text).

No comments: