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.

Purpose

I have long been fascinated by Fermat's Last Theorem and greatly excited by Andrew Wiles' proof. I have wanted for a long time to explore the history of the problem and also the amazing proof by Wiles.

Just this week, for example, I learned that another mathematician is about to provide a very significant result that builds on the work done by Wiles. For those interested this mathematician, Chandrashekhar Khare, has a home page here:http://www.math.utah.edu/~shekhar/. After I get to Wiles' proof, I will attempt to also analyze Khare's result.

This blog will be an effort to trace the history of Fermat's Last Theorem and work through the details of the second version of Wiles's proof. Fermat has been called the Prince of Amateurs (Bell, Men of Mathematics) so I hope it is appropriate that this blog is also run by a mathematical amateur.

My goal in this blog is to present a set of proofs in a style very much like Euclid. Each proof presented will either rely on a previous blog or rely on a set of identified postulates. I am not a professional mathematician so I may make mistakes. I am hopeful that other participants will provide useful comments to keep the quality of this blog up.

The solution to Fermat's Last Theorem embodies a large part of the history of mathematics including many of its major results and many of its most famous persons. This blog will start with Fermat. For the most part, it will follow a historical flow, going from Fermat to Euler to Gauss to Kummer, etc.

It would be a shame to trace the history of the proof without also looking at the lives of the mathematicians involved. Some blog topics will explore the life and achievements of the participants of this story.

I look forward to people adding their comments and corrections to the biographies and the proofs. I think that this is one of the most exciting stories in all of mathematics and a blog is a great way to explore it.

Popular Books on Fermat's Last Theorem

Fermat's Last Theorem is one of the most famous math problems of all time. When Andrew Wiles proposed that he had a solution, it was front page news.

There are numerous books which provide a general introduction to the problem and the history of its solution. The most popular one seems to be Simon Singh's Fermat's Enigma. Simon Singh also has a web site.

Additionally, PBS dedicated an entire episode of Nova to the investigation of the theorem and its solution by Andrew Wiles. The Nova episode is called The Proof. PBS has a web site.

The popular books and the Nova episode provide great background on the story and the significance of Wiles' proof. Unfortunately, they provide only a high level coverage of the proof's major ideas.

Many books have claimed to provide details for amateurs. I find all of these books to be very difficult. They are great for more advanced math students, but for most others, they can be quite a challenge.

• Fermat's Last Theorem for Amateurs - Explores the more elementary proofs.
• Notes on Fermat's Last Theorem - Great snippets. More a set of hints than actual proofs.
• Fermat's Last Theorem: A Genetic Introduction - Very in depth. Goes up to Kummer.
• Algebraic Number Theory and Fermat's Last Theorem - Very in depth. Broad coverage of Algebraic Number Theory.

In this blog, I will try to present details that bridge the gap between these works and Wiles'proof.