Fermat’s last theorem

Fermat’s last theorem

1

Fermat’s Last Theorem

Fermat’s last

theorem

was put forth by Pierre de Fermat around 1630. It states that the

Diophantine equation

(

a

,

b

,

c

,

n

∈

ℕ

)

a

n

+

b

n

=

c

n

has no non-zero solutions for

n

>

2

.

2

History

Fermat’s last theorem was actually a conjecture and remained unproved for over 300 years. It was finally proven in 1994 by Andrew Wiles, an English mathematician working at Princeton. It was always called a “theorem”, due to Fermat’s uncanny ability to propose true conjectures. Originally the statement was discovered by Fermat’s son Clement-Samuel among margin notes that Fermat had made in his copy of Diophantus’

Arithmetica

. Fermat followed the statement of the conjecture with the infamous teaser:

“I have discovered a truly remarkable proof which this margin is too small to contain”

Over the years, Fermat’s last theorem was proven for various sub-cases which required specific values of

n

, but no direct progress was made along these lines towards a general proof. These proofs were bittersweet victories, as each one still left an

infinite

number of cases unproved. Among the big names who took a crack at the theorem are Euler, Gauss, Germaine, Cauchy, Dirichlet, and Legendre.

The theorem finally began to yield to direct attack in the 20th century.

3

The Proof

In 1982 Gerhard Frey conjectured that if FLT has a solution (

a

,

b

,

c

,

n

), then the

elliptic curve

defined by

y

2

=

x

⁢

(

x

-

a

n

)

⁢

(

x

+

b

n

)

is

semistable

, but not modular. The above equation is known as Frey’s equation, or the

Frey curve

. Ribet proved this conjecture in 1986.

The

Taniyama-Shimura conjecture

, which appeared in an early form in 1955, says that all elliptic curves are modular. If in fact this conjecture were to be proven in the semistable case, then it would follow that the Frey equation would be semistable

and

modular, hence FLT could have no solutions.

After a flawed attempt in 1993, Wiles along with Richard Taylor successfully proved the semistable case of the Taniyama-Shimura conjecture in 1994, hence proving Fermat’s last theorem. The proof appears in the May 1995

Annals of Mathematics

, Vol. 151, No. 3. It is 129 pages.

4

Speculation

Wiles’ proof rests upon the work of hundreds of mathematicians and the mathematics created up to and including the 20th century. We cannot imagine how Fermat’s last theorem could be proved without these advanced mathematical tools, which include group theory and

Galois theory

, the theory of

modular forms

, Riemannian topology, and the theory of elliptic equations.

Could Fermat, then, have possibly had a proof to his own conjecture, in the year 1630? It doesn’t seem likely, given the requisite mathematics behind the proof as we know it. Assuming Fermat’s teaser was truthful, and Fermat was not in error, this apparent

paradox

has led some to jokingly

attribute

supernatural abilities to Fermat.

A more interesting possibility is that there is yet another proof, which is

elementary

and utilizes no more knowledge than Fermat had available in his day.

Most mathematicians, however, think that Fermat was just in error. It is also possible that he realized later that he didn’t have a solution, but of course did not amend the margin notes where he wrote his tantalizing statement.

Still, we cannot rule out the

existence

of a simpler proof, so for some, the search continues…

References

1

http://www-groups.dcs.st-andrews.ac.uk/ history/HistTopics/Fermat's_last_theorem.html

Fermat’s Last Theorem, by J J O’Connor and E F Robertson.

2

http://fermat.workjoke.com

Fermat’s Last Theorem, web site by David Shay.

3

http://www.simonsingh.com/fermat.htm

Fermat’s Enigma (offline book), by Simon Singh.

Title

Fermat’s last theorem

Canonical name

FermatsLastTheorem

Date of creation

2013-03-22 12:14:24

Last modified on

2013-03-22 12:14:24

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

15

Author

akrowne (2)

Entry type

Theorem

Classification

msc 11F80

Classification

msc 14H52

Classification

msc 11D41

Synonym

FLT

Related topic

TaniyamaShimuraConecture

Related topic

ArithmeticOfEllipticCurves

Defines

Frey curve

Defines

Frey’s equation

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