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Fermat's Last Theorem and more can be proved more simply

March 4, 2013
Case Western Reserve University
Mathematicians have shown Fermat's Last Theorem can be proved using only a small portion of Grothendieck's work. Specifically, the theorem can be justified using "finite order arithmetic."

Fermat's Last Theorem -- the idea that a certain simple equation had no solutions -- went unsolved for nearly 350 years until Oxford mathematician Andrew Wiles created a proof in 1995. Now, Case Western Reserve University's Colin McLarty has shown the theorem can be proved more simply.

The theorem is called Pierre de Fermat's last because, of his many conjectures, it was the last and longest to be unverified.

In 1630, Fermat wrote in the margin of an old Greek mathematics book that he could demonstrate that no integers (whole numbers) can make the equation xn + yn = zn true if n is greater than 2.

He also wrote that he didn't have space in the margin to show the proof. Whether Fermat could prove his theorem or not is up to debate, but the problem became the most famous in mathematics. Generation after generation of mathematicians tried and failed to find a proof.

So, when Wiles broke through in 1995, "It was just shocking to a lot of us that it could be proved," McLarty, said. "And we thought, 'Now what?' There was no new most famous problem."

McLarty is a Case Western Reserve philosophy professor who specializes in logic and earned his undergraduate degree in mathematics. He hasn't developed a proof for Fermat, but has shown that the theorem can be proved with much less set theory than Wiles used.

Wiles relied on his own deep insight into numbers and works of others -- including Alexander Grothendieck -- to devise his 110-page proof and subsequent corrections.

Grothendieck revolutionized numbers theory, rebuilding algebraic geometry in the 1960s and 1970s. He used strong assumptions to support abstract ideas, including the idea of the existence of a universe of sets so large that standard set theory cannot prove they exist. Standard set theory is composed of the most commonly used principles, or axioms, that mathematicians use.

McLarty calls Grothendieck's work "a toolkit," and showed, at the Joint Mathematics Meetings in San Diego in January, that only a small portion is needed to prove Fermat's Last Theorem.

"Most number theorists are like race car drivers. They get the best out of the car but they don't build the whole car," McLarty said. "Grothendieck created a toolkit to build cars from scratch."

"Where Grothendieck used strong set theory I've shown he could do with only a fraction of it," McLarty said. "I use finite-order arithmetic, where all sets are built from numbers in just a few steps.

"You don't need sets of sets of numbers, which Grothendieck used in his toolkit and Andrew Wiles used to prove the theorem in the 90s."

McLarty showed that all of Grothendieck's ideas, even the most abstract, can be justified using very little set theory -- much less than standard set theory. Specifically, they can be justified using "finite order arithmetic." This uses numbers and sets of numbers and set of those and so on, but much less than standard set theory.

"I appreciate the wholeness of the foundation Grothedieck created," McLarty said. " I want to take the whole thing and make it more usable to practicing mathematicians."

Mathematician Harvey Friedman, who famously earned his undergraduate, master's and PhD from MIT in three years and began teaching at Stanford University at age 18, calls the work a "clarifying first step," ScienceNews reported. Friedman, now an emeritus mathematics professor at Ohio State University, calls for McLarty's work to be extended to see if the theorem can be proved by numbers alone, with no sets involved.

"Fermat's Last Theorem is just about numbers, so it seems like we ought to be able to prove it by just talking about numbers," McLarty said. "I believe that can be done, but it will require many new insights into numbers. It will be very hard. Harvey sees my work as a preliminary step to that, and I agree it is."

McLarty will talk more about that specific result at the Association for Symbolic Logic North American Annual Meeting in Waterloo, Ontario, May 8-11.

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