Sunday, May 01, 2005

The Problem

For this blog, I assume that everyone is familiar with exponents. If you are not, here is an introduction.

Fermat's Last Theorem states for the equation: xn + yn= zn, there are no whole number solutions where x * y * z ≠ 0 and n > 2.

If x * y * z = 0, then it is easy to find a solution. For example (5)n + (0)n = (5)n.

Likewise, if we consider real numbers, then the solution is straight-forward algebra:
z = (xn + yn)(1/n).

Finally, if n = 2, then we have the Pythagorean Theorem a2 + b2 = c2. This is solveable by any Pythagorean Triple such as 3,4,5 (32 + 42 = 52) .

I think that this is the real appeal of the problem. It is easily stated and on its surface looks like it shouldn't be too difficult to resolve one way or the other.

Pierre de Fermat rarely published any of his results. He prefered to describe the problem and claim that he had found a solution. This has made the problem even more appealing: did Fermat actually have a proof?

The theorem itself became public without proof in 1670 when Fermat's son, Clement-Samuel published his father's notes. Unfortunately, Fermat was not around to explain his famous theorem because he had died in 1665. Instead, the reader was left with the famous statement of the problem:
"It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum oftwo like powers."
And this very mysterious statement about the proof:
"I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." (both quotes are from Fermat's Engima)
For over 350 years, this problem remained unsolved. Many of the greatest mathematicians were able to make progress on the problem including Leonhard Euler, Carl Friedrich Gauss, and Ernst Kummer but none of these great minds offered a solution.

The solution had to wait until 1995.

6 comments:

Justin said...

You may appreciate seeing this famous quote in the original Latin:

"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet."

I will translate this more literally than I would normally, just so you can see how the Latin works:

"But to divide a cube into two cubes, or a doublesquare into two doublesquare and generally no power up to infinity from beyond the square into two of the same name, is not permissible. Of which thing I have of course uncovered a wonderful proof. The smallness of the margin would not be able to contain it."

Especially interesting is the the first meaning of fas is permissible in a religious sense. Something like "in accordance with divine law." It is possible to use it in a broader sense, but I enjoyed the implication that finding a solution to x^3+y^3=z^3 would be a sin! ;)

Larry Freeman said...

Hi Justin,

Thanks very much for the details on the translation!

-Larry

Madhu Sudhan said...

Sir,
I guess i have found a simpler proof for fermat's last theorem... I wish it to be recognised by the right people.... I think that you know the right contacts so can you please help me??????

Larry Freeman said...

Hi Madhu,

I'm just an amateur. So, I don't have any contacts.

My recommendation is to write up, post it somewhere public, and get feedback. Most likely, you have made some significant mistakes. Even Andrew Wiles made a giant mistake.

If you have thought it through deeply enough, you will be able to recover from some of the mistakes. If you believe your argument still holds, then try to publish it on http://arxiv.org/.

If you post it somewhere public, you can link to it in the comments on this blog.

Cheers,

-Larry

Rashid ullah khan hunxai said...

sir,
I was busy with the proof of Phythagoras theorem,I have some Differential equations with the help of which i formulate proof of Fermat's theorem i want you to check my results.

Rashid ullah khan hunxai said...

sir,
I was busy with the proof of Phythagoras theorem,I have some Differential equations with the help of which i formulate proof of Fermat's theorem i want you to check my results.