Fermat's Last Theorem?

Question

Why does this theorem have this name if, in fact, it was never proved by Fermat (although Fermat was, indeed, a great mathematician)? For what I know, it was proved by Andrew Wiles in 1996, I guess.

Answer 1

Pierre de Fermat created the Last Theorem while studying "Arithmetica", an ancient greek text written in about AD 250 by Diophantus of Alexandria. The page of Arithmetica which inspired Fermat discussed various aspects of Pythagoras’ Theorem, which states that "in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares on the other two sides". In math words:
x² + y² = z².

Fermat was interested in whole number solutions to Pythagoras’ equation, such that x, y, and z could be any whole number, except zero. Fermat must have been bored with such a tried and tested equation, and as a result he considered a slightly mutated version of the equation:
x³ + y³ = z³

Surprisingly, the frenchman came to the conclusion that among the infinity of numbers there were none that fitted this new equation. Whereas Pythagoras’ equation had infinite number of possible solutions, Fermat claimed that his equation was insoluble.

Fermat went even further, believing that if the power of the equation is increased further, then these equations would also have no solutions, and stated the theorem:

" There are no solutions in integers x, y, z for n > 2 of the equation xⁿ + yⁿ = zⁿ "

While editing the book "Arithmetica" he wrote:
"Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere...
.....Cuius rei demonstrationem mirabilem sane detexi hanc marginis exguitas non caperet."

Or, for those who can't understand this baddly spoken portuguese (I'm kidding, of course. Indeed, the opposite is true ... portuguese is baddly spoken latim - lol :)

"It is impossible for a cube to be written as a sum of two cubes, or a fouth power to be written as the sum of two forth powers, or, in general, for any number which is a power greater than the second to be written as the sum of two like powers...
....I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain."

Fermat believed he could prove his theorem, but he never committed his proof to paper. It is believed that the creation and proof of the Last Theorem happened in about 1637, but it was not until after Fermat’s death in 1665 that his marginal note came to light. His son, Clément-Samuel, discovered the casual jotting along with many others, all hinting at theorems, but at best giving only a glimpse of the underlying proof. Then in 1670 he published Diophantus’ Arithmetica Containing Observations by P. de Fermat, which contained Diophantus’ original text interspersed by Fermat’s notes.

What a puzzle! It fooled mathematicians for centuries ... even Leonhard Euler, the greatest mathematician of the 19th century, was defeated.

A bit of humor: in the 1980s, a piece of graffiti appeared on New York’s Eighth Street subway station:
"xⁿ + yⁿ = zⁿ , no solutions. I have discovered a truly marvellous proof of this, but I can’t write it down because my train is coming"

Fermat's Last Theorem was finally proved by Andrew Wiles in 1993, but a minor error forced him back to his researches. He announced his final proof in 1995.

It took 358 years to fill in that margin, but Wiles' proof was certainly not what Fermat had in mind: the proof has over 200 pages and uses a very sophisticated set of new (last 30 years or so) developments in the theory of elliptic curves.

I recommend the book listed bellow. It's really amazing !

Additional info: about 2/3 of the 600,000 words (approx.) in english language came from latim.
 
 

Answer 2

For what I know, he stated it as a "conjecture" and claimed to have proves it although there was no documentation of the proof. Some believe that the complexity of the proof make it more likely that he did not in fact prove it in his time only claimed to have done so(unless there is an easier solution that everyone else missed). I think they should call it Fermat-Wiles since Wiles
was the one who finally proved it.

Answer 3

It's gains the name because it was one of the most infamous theorems in history. It was also the last of Fermat's theorems to be solved by Andrew Wiles in 1994. All his other theorems were proven by himself or other mathematicians before 1994.

Answer 4

Fermat claimed to have proved it. He mentioned his proof in passing in his notes. He never published a proof. Indeed the complexity of the proof by Wiles makes a Fermat proof highly unlikely. Likely he was either mistaken in his notes or being deceptive. But a mysterious proof, that is alluded to but never discovered!? The romance of it all contributed greatly to its association with Fermat.

Answer 5

Fermat stated the equation.

Answer 6

The wikipedia article answers your question slightly. It was his theory and he proved the first part. Since it was his "theory" it was named after him.

http://en.wikipedia.org/wiki/Fermat's_last_theorem

Answer 7

x^n+y^n=z^n, for n./or=3, there is no solution.