Fermat's Last Theorem - Exponent Three

[Brimley]

Hello Physics Forums,

I read around and saw a few examples for Fermat's last theorem for exponents 1 and 2, but I was wondering if this can be proven for exponent 3. That is:

Proof that IF $x^3+y^3=z^3$, where $x,y,$ and $z$ are rational integers, then $x, y,$ or $z$ is $0$.

Can this be done?

 

[epsi00]

Professor Google may have the answer for you. He never sleeps.

 

[robert Ihnot]

Fermat's Last Theorem does not apply for exponents 1 and 2. The proof for n=3 was first attempted by Euler. The proof used by Hardy and Wright, An Introduction to the Theory of Numbers, involves the Eisenstein numbers, i.e. a+bu, where u represents the cube root of 1. This involves a certain amount of difficulity.

 

[robert Ihnot]

A further historical note: Fermat is assumed to have written his famous note about his Last Theorem in the margin around 1673. (He did not publish on the matter.) But it was not until 1770, more than 130 years later, that Euler came up with his proof, which was correct but contained an omission.

So one has to see how difficult this problem would prove for amateurs knowing little math. In fact, in 200 years the only cases proven were 3,5,7.

 

[mathwonk]

i always thought 4 was the easiest and hence presumably first case. perhaps you are thinking of prime exponents, but since the case 2 has solutions, you still seem to need to do the case of exponent 4.

 

[robert Ihnot]

The case for n=4, having no solutions, was given by Fermat. Though this can be somewhat disputed since he showed exactly that a right triangle cannot have the same area as a square, which is equivalent. http://fermatslasttheorem.blogspot.com/2005/05/fermats-last-theorem-n-4.html

 

[mathwonk]

Robert: thanks for the history. my post was inspired by this quote in yours:

" In fact, in 200 years the only cases proven were 3,5,7."