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What is the Last Theorem?

Fermat’s Last Theorem is the most notorious problem in the history of mathematics and surrounding it is one of the greatest stories imaginable. It dominated my own life for four years, because I made a TV documentary, wrote a book and then lectured on the subject. Getting involved in Fermat’s mischievous conundrum set me on the path towards being an author and ignited an interest in mathematics that has continued ever since. As a physicist, I was always interested in mathematics as a tool for studying the universe, but learning about Fermat’s Last Theorem taught me to love mathematics for its own sake.

What is the Last Theorem?

Pierre de Fermat created the Last Theorem while studying Arithmetica, an ancient Greek text written in about AD 250 by Diophantus of Alexandria. The page of Arithmetica which inspired Fermat discussed various aspects of Pythagoras’ Theorem, which states that:

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

In other words (or rather symbols):

x2 + y2 = z2

where z is the length of the hypotenuse, the longest side, and x and y are the lengths of the other two sides.

Pythagoras’ Theorem is not just a nice idea, or a notion that seems to work for most right-angled triangles. It is always true and mathematicians can prove this.
Fermat was interested in whole number solutions to Pythagoras’ equation, such that x, y, and z could be any whole number, except zero. For example:

32 + 42 = 52 (i.e. 9 + 16 = 25)
52 + 122 = 132 (i.e. 25 + 144 = 169)

The numbers (3, 4, 5) or (5, 12, 13) are known as Pythahorean triples, and such triples have been studied for thousands of years. Indeed, ancient Babylonian tablets list Pythagorean triples.

There are an infinite number of Pythogorean triples. This can be demonstrated by looking at looking at the difference between successive square numbers. You can see that every odd number is the difference between two squares. Therefore every square odd number is the difference between two squares. There are an infinite number of square odd numbers, so there must be an infinite number of Pythagorean triples.

22 – 12 = 4 – 1 = 3
32 – 22 = 9 – 4 = 5
42 – 32 = 16 – 9 = 7
52 – 42 = 25 – 16 = 9
62 – 52 = 36 – 25 = 11
72 – 62 = 49 – 36 = 13
82 – 72 = 64 – 49 = 15

Fermat must have been bored with such a tried and tested equation, and as a result he considered a slightly mutated version of the equation:

x3 + y3 = z3

Surprisingly, the Frenchman came to the conclusion that among the infinity of numbers there were none that fitted this new equation, which is said to be cubic or to the third power. Whereas Pythagoras’ equation had many possible solutions, Fermat claimed that his equation was insoluble.

The Pythagorean equation and the cubic equation can be visualised in 2 or 3 dimensions. In two dimensions it is easy to add the tiles of one square to another square to create a third bigger square, as shown below:

In three dimensions it seems impossible to add the blocks of one cube to another cube to create a third bigger cube. This can be seen below:

Fermat went even further, believing that if the power of the equation is increased further, then these equations would also have no solutions:

x3 + y3 = z3
x4 + y4 = z4
x5 + y5 = z5
x6 + y6 = z6

The mathematical short-hand for this family of insoluble equations is:

xn + yn = zn , where n is any number greater than 2.

According to Fermat, none of these equations could be solved, and he noted this in the margin of his Arithmetica. To back up his theorem he had developed an argument or mathematical proof, and following the first marginal note he scribbled the most tantalising comment in the history of mathematics:

“Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere…

…..Cuius rei demonstrationem mirabilem sane detexi hanc marginis exguitas non caperet.”

or to put it another way..

“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 the sum of two like powers…

….I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.”

Fermat believed he could prove his theorem, but he never committed his proof to paper. It is believed that the creation and proof of the Last Theorem happened in about 1637, but it was not until after Fermat’s death in 1665 that his marginal note came to light. His son, Clément-Samuel, discovered the casual jotting along with many others, all hinting at theorems, but at best giving only a glimpse of the underlying proof. Then in 1670 he published Diophantus’ Arithmetica Containing Observations by P. de Fermat, which contained Diophantus’ original text interspersed by Fermat’s notes.

Now the race was on to rediscover Fermat’s proof. Trial and error showed that Fermat’s Last Theorem seemed to be true, because nobody could find three whole number solutions. But nobody could be sure that no such solutions existed. Mathematicians would only be happy if they could find a solid proof, a reasoned argument, something that would unequivocally show that the theorem was true.

Fermat’s Last Theorem became the most notorious problem in mathematics. The more that mathematicians tried, the more they failed, and the more desirable a solution became. The Last Theorem was a source of frustration, but it also had a lighter side. In the 1980s a piece of graffiti appeared on New York’s Eighth Street subway station.

xn + yn = zn , no solutions.
I have discovered a truly marvellous proof of this,
but I can’t write it down because my train is coming.