Posted by: gupta2 | October 30, 2008

## Fermat’s Last Theorem

Okay so here’s a theorem that relates a little to what we have been doing in class with rings.

Fermat’s last theorem states: “It is impossible to separate any power higher than the second into two like powers” or “If an integer n is greater than 2, then the equation an + bn = cn has no solutions in non-zero integers a,b, and c.”

Despite the fact that this theorem looks a lot like the Pythagorean theorem (that’s when n=2), it is many times more difficult to prove. The similarity of the answer actual to the Pythagorean theorem has spawned many incorrect proofs but in 1995 Andrew Wiles published a proof for it. The name Fermat’s Last Theorem comes from the fact that this conjecture was the last of his theorems to be proven or disproven.

Fermat’s Theorem only has to be proven for prime numbers greater than 2 and for 4. The last part is because any even n greater than two can just be written in the form:

where p=2.
Although that part of the proof could be shown to be untrue. The part Wiles’s did was not so easy and used the modular properties of elliptical curves: “all elliptical curves can be parameterized by a rational map with integer coefficients using the classical modular curve” (Taniyama-Shimura) To prove Fermat’s theorem, Wiles used the semistable elliptical curve: y2 = x(x − ap)(x + bp) to show that when substituted into Fermat’s equation it did not produce modular results.
The overall proof is very difficult and makes use of algebraic geometry and number theory and has affected all of mathematics. Here is a link to the paper, if you’re up to it:

When Wile’s proved this theorem there was actually a large uproar in the math and general community. It was actually the first time that major media coverage followed a mathematical proof and it even had its own PBS Nova special. However, the proof was not without some controversy. After working on the proof for seven years in complete secrecy and without any outside help, Wile’s announced at the Isaac Newton Institute for Mathematical Sciences on June 21, 22, and 23 that he had a proof of the Taniyama-Shimura conjecture and Fermat’s Last Theorem. However, Wiles and his collegue Kick Katz found a gap in the proof that actually bounded the order of a certain group. In 1994, Wiles and his student, Taylor, repaired the proof under the watch of the media and the math community. They actually went on and proved the Taniyama-Shimura conjecture which is also known as the modularity theorem which can actually be applied to any elliptic curve and not just the semistable curves Wile’s need to prove Fermat’s theorem.

After finding out more about Fermat’s theorem, it is highly unlikely that Fermat with his element but important mathematical resources could have found an elemental proof of this theorem as he claimed in the margin of a paper. It is more likely that Fermat found a proof under certain conditions. If Fermat in fact did find a general proof then in his 60 year lifespan in 1600s he was able to outsmart everyone for the 350 years that followed him. Given that some of these people had access to computers and other advanced computational equipment, it’s safe to say that Fermat was truly impressive.