Featured Research

from universities, journals, and other organizations

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

Date:
March 4, 2013
Source:
Case Western Reserve University
Summary:
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.


Story Source:

The above story is based on materials provided by Case Western Reserve University. Note: Materials may be edited for content and length.


Cite This Page:

Case Western Reserve University. "Fermat's Last Theorem and more can be proved more simply." ScienceDaily. ScienceDaily, 4 March 2013. <www.sciencedaily.com/releases/2013/03/130304105652.htm>.
Case Western Reserve University. (2013, March 4). Fermat's Last Theorem and more can be proved more simply. ScienceDaily. Retrieved July 29, 2014 from www.sciencedaily.com/releases/2013/03/130304105652.htm
Case Western Reserve University. "Fermat's Last Theorem and more can be proved more simply." ScienceDaily. www.sciencedaily.com/releases/2013/03/130304105652.htm (accessed July 29, 2014).

Share This




More Computers & Math News

Tuesday, July 29, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

It's Not Just Facebook: OKCupid Experiments With Users Too

It's Not Just Facebook: OKCupid Experiments With Users Too

Newsy (July 29, 2014) If you've been looking for love online, there's a chance somebody has been looking at how you're looking. Video provided by Newsy
Powered by NewsLook.com
Why Facebook Wants You To Download Its Messenger App

Why Facebook Wants You To Download Its Messenger App

Newsy (July 29, 2014) Facebook will start requiring users to download a separate Messenger application if they wish to continue using Facebook for mobile messaging. Video provided by Newsy
Powered by NewsLook.com
Teen's Phone Ignites Under Her Pillow; How Real Is The Risk?

Teen's Phone Ignites Under Her Pillow; How Real Is The Risk?

Newsy (July 28, 2014) A Texas teen's Samsung phone apparently ignited while she slept, but what was the real problem here? Video provided by Newsy
Powered by NewsLook.com
Zillow Snaps Up Web Real Estate With Trulia Deal

Zillow Snaps Up Web Real Estate With Trulia Deal

Reuters - US Online Video (July 28, 2014) Zillow's decision to buy rival Trulia is just one step in a continuing string of acquisitions, and Zillow CEO Spencer Rascoff is already thinking about his next big deal. Bobbi Rebell reports. Video provided by Reuters
Powered by NewsLook.com

Search ScienceDaily

Number of stories in archives: 140,361

Find with keyword(s):
Enter a keyword or phrase to search ScienceDaily for related topics and research stories.

Save/Print:
Share:

Breaking News:
from the past week

In Other News

... from NewsDaily.com

Science News

Health News

Environment News

Technology News



Save/Print:
Share:

Free Subscriptions


Get the latest science news with ScienceDaily's free email newsletters, updated daily and weekly. Or view hourly updated newsfeeds in your RSS reader:

Get Social & Mobile


Keep up to date with the latest news from ScienceDaily via social networks and mobile apps:

Have Feedback?


Tell us what you think of ScienceDaily -- we welcome both positive and negative comments. Have any problems using the site? Questions?
Mobile: iPhone Android Web
Follow: Facebook Twitter Google+
Subscribe: RSS Feeds Email Newsletters
Latest Headlines Health & Medicine Mind & Brain Space & Time Matter & Energy Computers & Math Plants & Animals Earth & Climate Fossils & Ruins