# SOU talk explains Fermat's Last Theorem

For centuries, one of the biggest puzzles in mathematics was the fact that no one could offer proof to substantiate Fermat's Last Theorem, which was put forth by Pierre de Fermat in 1637. It even made the Guinness World Records' list of most difficult unsolved mathematical problems.

But those days are gone. Several late 20th-century math wizards cracked the problem and one of them, Dr. Ken Ribet of the University of California at Berkeley, will tell how it was done.

Ribet will give a talk, "Fermat's Last Theorem and the Modularity of Elliptic Curves," at 4 p.m. today in Room 118 in the Science Building at Southern Oregon University. It is free and open to the public.

Ribet in 1990 "constructed a logical argument by making a complete link" between the theorem and problems involving elliptical curves, he said in a phone interview, enabling Oxford University Professor Andrew Wiles in 1993 to finally prove the theorem, a feat winning Wiles worldwide notoriety in the news, books and even a "NOVA" documentary.

The complex and esoteric theorem says no three positive integers can be used in the equation an + bn = cn for any integer where "n" is greater than two.

In other words, you can square and add a and b and get a positive integer, but you can't cube (or use any larger exponent than two) and add them and get a positive integer.

"It was a puzzle in mathematics," Ribet said, noting that serious professionals in the field used that term in a pejorative way because it was a "deceptively simple" statement that no one could resolve for three-plus centuries.

As an example, Ribet said 5-squared plus 12-squared equals 13-squared, or 169, but if the exponent is bigger you can't get "perfect squares."

"Fermat posed this challenge to a friend and said he had proof but there wasn't enough room in the margin of the letter to state it," says SOU mathematics Professor Kemble Yates. "But now, no one believes he had the proof. It was a challenge through the centuries, the Holy Grail of mathematics, and in 1993, Wiles claimed it was true and it is.

"His proof was of such richness and depth. It was so incredible. That's why Fermat probably didn't have it."

Yates adds that Ribet came up with a "key result (proof for the 'epsilon conjecture' of French mathematician Jean-Pierre Serre) needed to complete the proof. The gravity of Ribet's work is clear in that he got his own Wikipedia entry for it."

Ribet notes that the Fermat problem "became the single most famous mathematics problem of the 20th century, as people attempted to find ingenious arguments, which gave rise to countertheories, so by the time I was a grad student, the professionals had lost interest and they thought only amateurs would fool with it. It seemed an isolated puzzle and maybe the proof could never be found."

Working with numbers theory about 40 years ago, German mathematician Gerhard Frey constructed the Frye Curve, which he put forth to prove the Fermat issue, a key step that Ribet proved — and the Ribet Theorem, he said, became a key link for the Wiles proof.

Ribet will speak to SOU mathematicians in a 10:30 a.m. lecture today called "Elliptical Curves," in Taylor Hall rooms 29 and 30.

Using Sage software, Ribet says he will "give the audience a taste" of the proof and the software "which should be in every mathematician's took kit."

The talks, he said, will be "non-technical." Despite all the work and celebration about proving the theorem, Ribet says he knows of no "real-world application" for it, but adds that "lots of structures and arguments have been very useful in later problems."

John Darling is a freelance writer living in Ashland. E-mail him at jdarling@jeffnet.org.