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Fermat's Last Theorem

 

Or, more precisely, Fermat's conjecture (A conjecture is a theorem before it has been proved). This conjecture became one of the "holy grails" of mathematics, and was unsolved for hundreds of years.

Fermat.jpg (32434 bytes)

Introduction

Let's start by proving something similar, though not related. Pythagorus' Theorum, a specific case of the Cosine Rule, can be used to ascertain the hypotonuse of a right angled triangle:

pythag.jpg (12245 bytes)

The geometric proof for this is quite simple.  If a smaller square is placed inside a larger square, we get the following shape (Note the pronumerals):

pythag2.jpg (10499 bytes)

The formula for the TOTAL area of the shape can be obtained one of two ways.

1. By Using the large square:

pythag3.jpg (4946 bytes)

2. Or by using the smaller square, plus the area of the four triangles:

pythag4.jpg (3848 bytes)

If we equate these two equations, we get:

pythag5.jpg (5507 bytes)

Simple, easy proof, but nothing to do with Fermat's Last Theorem.

 

The Theorem Itself

Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2.

In easy-speak, no natural numbers will make that equation work apart from 0, 1 and 2. He then went on to say that he has proved it, but couldn't be bothered writing it as it wouldn not fit in the margin. Whether or not he did prove it will always remain a mystery, however the scale of the proof required would infer that there is a very slim chance that he got anywhere near it.

Infact, over 1000 false proofs were published between 1908 and 1912.

 

Why is it so hard to prove?

You can't just insert numbers and try it. A proof such as this one means that it must hold true for ALL numbers. If you found three numbers that made it work, then, fine, the conjecture is disproven. However, simply computing to prove the equation IS true cannot be done. If you tried every number up to 1000, there are still more numbers to be tried. If you got a computer to try every number up to 100 000 000 000 000 000 000 000, there are still more possibilities, and you are no closer to solving the problem.

Andrew Wiles

To cut a long story short, The conjecture was not proven until the 1990's, When Andrew Wiles locked himself in a bedroom for seven years and put his head down. His work was based on the work of many who went before him, and there is not enough space here to cover all the mathematicians who contributed, however the important developments will be listed here:

  • 1955 - Yataka Taniyama worked on eliptic curves, of the equation y2 = x3 +ax + b. Finally, the Shimura-Taniyama-Weil Conjecture was established.

  • 1986 - This conjecture was linked to Fermat's Last Theorem, showing that the theorem related to the fundemental properties of 3D space as we know it... WOW!

  • 1993 - Andrew Wiles, of the U.K., claims to have proved the conjecture, by proving the part of the Shimura-Taniyama-Weil Conjecture that was vital to proving the Theorem.

  • December, 1993 - After speculation, Andrew Wiles releases a statement admitting there are faults in the proof "The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures." A chain is only as strong as its weakest link - The proof was therefore worthless.

  • 1994 - Wiles works with Richard Taylor to patch the holes in the proof. The realise that the Flach Method could not be patched, and therefore they had to find an alternative way to prove it. They did this, using an Euler Method that Wiles had used previously, essentially simplifying the proof.

  • 1995 - It is accepted that, whilst no proof of the complexity can be taken yet as truth, There is little doubt that the theorem had been proved.

Thanx to this site:

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html

If anyone would like to add anything, or know more, email me stewy6@hotmail.com.