For the DU Mathematicians - Elegant New Approach To 350-Year-Old Math Riddle

[malaise]

http://www.redorbit.com/news/science/1112796238/fermats-last-theorem-gets-simplified-030413/
<snip>
In 1637, the French lawyer and part-time mathematician Pierre de Fermat put forward a simple and elegant numerical riddle that would puzzle and confound math geeks for 358 years. Known as Fermat’s Last Theorem, or simply Fermat’s conjecture, the theorem states no whole, positive numbers can make the equation xn + yn = zn true when ‘n’ is greater than 2.

Using a roundabout, backdoor approach that involved dizzyingly complex number theory, the Oxford mathematician Andrew Wiles demonstrated conclusively in 1995 that Monsieur de Fermat was, in fact, correct: The equation is unsolvable.

Now, however, Colin McLarty, a professor of philosophy and mathematics at Case Western Reserve University claims there’s a far simpler way to prove Fermat’s theorem – one that doesn’t involve complex mathematical wizardry with names like “modularity theory” and “epsilon conjectures.”

McLarty demonstrated even the most complex and abstract of Grothendieck’s ideas can be justified using very little set theory. What is known as ‘standard set theory’ is simply the collection of the most commonly used principals, or axioms, used by practicing mathematicians. Grothendieck’s work included the notion of the existence of a universe of number sets so large standard set theory could not even prove they exist.

In McLarty’s vastly simplified approach to Fermat’s problem, he says all mathematicians need is basic ‘finite order arithmetic,’ which uses even fewer sets of numbers than standard set theory.

[ananda]

1. Well, between McLarty and Friedman ...

... there ought to be a next step coming around before too long.

[Progressive dog]

2. In Fermat's version n is a power, not a multiplier

That's in the original article too, but it's wrong. The equation that is written is easily solved.
If you can, could you correct this? Thanks

[DrDan]

3. exactly - thank you - any first year algebra student can solve what is written

[Princess Turandot]

4. ...

xⁿ + yⁿ = zⁿ

[Progressive dog]

5. Thanks it was bugging me. nt

[eppur_se_muova]

7. x[sup]n[/sup] + y[sup]n[/sup] = z[sup]n[/sup]

Use sub for subscripts (advice from a chemist ).

xⁿ + yⁿ = zⁿ

[bemildred]

6. Number theory, where the real nerds romp and play.

I think Mr. Friedman is right, they will get a purely numerical proof eventually, Fermat claimed he had one way back then.

Thanks for posting.

[SwissTony]

8. Hmmm...McLarty claims there's a far simpler way to prove Fermat's Theorem...

but doesn't present it. Does that mean

a) he has a proof but doesn't want to present it just yet
b) a simpler proof is possible in McLarty's opinion but he may or may not have found it

"McLarty says he has demonstrated the correctness of Fermat’s Last Theorem without a mathematical proof and with far less abstract and circuitous theory than that used by Wiles nearly two decades earlier" seems to imply he has found the proof.

"Friedman...called McLarty’s work a 'clarifying first step'” suggests he's possibly on the way but not there yet.

I'm not familiar with redOrbit, so it may be slightly misquoting McLarty or just exaggerating.

And, of course, we've had many "proofs" of FLT before. Only that of Wiles is still around.

[bhikkhu]

15. Oddly familiar that he wouldn't publish an actual proof with the claim

something like "I have assuredly found an admirable proof of this, but the margin is too narrow to contain it." :p

[adieu]

9. This is not a new idea

The traditional approach that most mathematicians had tried to prove FLT had been to look at the equation purely as a number theory question. You don't think it failed the best minds of the past 350 years because they decided to create the whole world of elliptic curves and algebraic geometry to try to solve it. Most people, professional and amateur mathematicians, tried solving it the "number theory" way.

In a way, they had succeeded for various numbers of "n". First proof fo n = 3 was done, not too difficult. Then 4, then 5, then, some advances as it was proven for all n < (some big number). Then that (some big number) grew even bigger. But, mathematically, it doesn't matter if it's just true for all n less than some huge honkin' number. The theorem has to be for all n > 2.

The problem was that the proof for n = (some number) doesn't extend well to other n's. It's like each n had to have it's own proof. Well, that can get tiresome.

So, I doubt McLarty has any more insight to the problem than the past 350 years' worth of top mathematicians.

[jimlup]

10. My guess remains that the "proof" Fermat conjectured

was probably incorrect or incomplete. It seems to me unlikely that a "simple" proof would not have been found given the application of significant brain power against it. The number of people with means who have thought seriously about the problem is at least an order of magnitude larger than the number who might have done so in Fermat's time.

[edhopper]

11. This is true

but it would still be very interesting to know what Fermat's was. And the fact that he was correct probably means it is intriguing at the least.

[jimlup]

13. Yeah, this is one of the most intriguing stories on the rich tradition of number theory

[edhopper]

14. The marginal notation

that launched centuries of study

[rurallib]

12. I think there is some brain power over at

Free Republic that could dig into this.

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(sarcasm for the humor impaired)

[malaise]

16. Best laugh of the day