Fermat and the Last Theorem

FERMAT AND THE LAST THEOREM

On the 20th of August 1601, a person was born in South-West France in a town called Beaumont-de-Lomange. That person, whose name was Pierre de Fermat, would grow up to become the originator of one of the world's most enduring puzzles.

(Statue of Fermat. Image from wikimedia commons)

The puzzle he invented can trace its origins way back before Fermat's birth to the sixth century BC. At that time, in Samos, there was a brilliant man by the name of Pythagorus. It was he who first used the word 'philosopher', which he defined as 'he [who] seeks to uncover the secrets of nature', pursuits he felt described the 'Pythagorean Brotherhood', a group comprising of six hundred followers of Pythagorus's teachings.

What Pythagorus is most famous for is his contribution to mathematics. For the purposes of this essay, only one of his discoveries need be explained, and that is 'Pythagoras' Theorem', which is expressed in the following way:

"In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides".

This can be expressed with the equation X^2+Y^2=Z^2.

You will notice that this theorem mentions squares. It talks about the 'square on the hypotenuse' and 'the sum of squares on the other two sides'. We can visualise the equation by taking one square made up of nine tiles, and another larger square made up of 16 tiles like so:

If we then take the tiles from the smaller square and arrange them around the outer edge of the larger square like so:

...you can see how the result is a square made up of 25 tiles. This is a visual demonstration of the fact that 3^2+4^2=5^2 or 9+16=25.

Now, what does all this have to do with that Frenchman, Pierre-de-Fermat? He wondered what would happen if, instead of finding 'whole number solutions' to square numbers (which, remember, can be visualised as taking two squares comprised of a certain number of tiles and recombining those tiles to make one larger square) and tried to accomplish the same feat with 'cubed numbers'. Visually speaking, this would be like taking two cubes comprised of smaller cubes, like these cubes constructed out of Minecraft blocks

....and then recombining them to form one, larger cube. So, let's say you have one cube comprised of 216 blocks (that's six times six times six) and another made up of 512 blocks (eight times eight times eight). What do you get when you recombine those blocks? Well, you get an object comprised of 728 blocks. But 728 is one digit short of the cubed number 729, so if you were to try to make a complete cube out of 728 smaller cubes, the result would look like a Rubick's cube with one cube missing.

Still, we have only attempted to make a single larger cube out of two cubes, one comprised of 216 blocks and the other made up of 512. But there are an infinite amount of cubed numbers, which means an infinite amount of 'cubes-made-out-of-smaller cubes' which you could try to recombine in order to make one, larger, complete cube-made-out-of cubes.

However, while finding whole number solutions to the equation x^2+y^2=z^2 is not too difficult, nobody had ever managed to find a perfect fit for the equation x^3+y^3=z^3. To put that in visual terms, anyone who has tried to take two cubes-made-out-of-smaller-cubes and recombined the building blocks of those cubes to produce one single cube, has either ended up with insufficient building blocks to complete the cube, or else has a cube and some spare building blocks left over.

Fermat is arguably most famous for apparently proving that you cannot take two Rubick's Cubes of differing size, disassemble them and then recombine their building blocks to make one larger cube. Fermat alleged that "it is impossible for a cube to be written as a sum of two cubes". Indeed, he went further than that, claiming "it is impossible...for any number which is a power greater than the second to be written as two like powers".

In other words, Fermat was claiming that there are no whole number solutions to the equation X^n+Y^n=Z^n where 'n' is any number greater than 2. What that means is that you apparently cannot take two four dimensional Rubick's objects (or fifth dimensional ones, sixth, and so on up to infinity) and recombine their component parts to make a single, complete, larger higher-dimensional Rubick's object.

But, really, this claim is not what made Fermat so famous. Rather, it was a remark he jotted down after having claimed no whole number solutions to X^n+Y^n=Z^n: "I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain".

From that day onward, the claim "there are no whole number solutions to the equation X^n+Y^n=Z^n where 'n' is any number greater than 2" has become known as 'Fermat's Last Theorem'. Strictly speaking, this should have been known as 'Fermat's Last Conjecture'. The reason why is because, in maths, you cannot just assert that you have proved something, you have to provide a logical argument that explains beyond all doubt that your conclusion is correct. For example, you cannot just assert that there are an infinite number of integers (that's 1, 2, 3...) nor can you say 'well I have counted to X and not reached the final integer yet, I reckon they go to infinity'. You have to present a logic argument, something like 'let's suppose there is a largest integer. Call it 'N'. What happens when you add 1 to 'N'? The result is always an integer larger than N. This proves the integers go on to infinity'.

Fermat, though, was something of a mischievous genius so he did not bother to write down the proof that showed, logically, that his claim "there are no whole number solutions to the equation X^n+Y^n=Z^n where 'n' is any number greater than 2" had to be correct. However, he was accomplished enough at maths for other mathematicians to take his claim that there was a way to prove 'Fermat's Last Theorem' is true seriously. Since the day he claimed to have found such a proof, his fellow mathematicians have attempted to 'rediscover' the logical proof of his Last Theorem'. For over three centuries, people tried and failed to prove what Fermat claimed to have proved way back in 1600s, including some of the most brilliant mathematicians and logicians ever to have lived.

How this theorem was eventually proved will be discussed in an upcoming essay.

REFERENCES

"Fermat's Last Theorem' by Simon Singh

BBC Horizon documentary on Andrew Wiles