In three dimensions it seems impossible to add the blocks of one cube to another cube to create a third bigger cube. 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: The Pythagorean equation and the cubic equation can be visualised in 2 or 3 dimensions. Whereas Pythagoras’ equation had many possible solutions, Fermat claimed that his equation was insoluble. 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. There are an infinite number of square odd numbers, so there must be an infinite number of Pythagorean triples.įermat must have been bored with such a tried and tested equation, and as a result he considered a slightly mutated version of the equation: Therefore every square odd number is the difference between two squares. You can see that every odd number is the difference between two squares. This can be demonstrated by looking at looking at the difference between successive square numbers. There are an infinite number of Pythogorean triples. Indeed, ancient Babylonian tablets list Pythagorean triples.

The numbers (3, 4, 5) or (5, 12, 13) are known as Pythahorean triples, and such triples have been studied for thousands of years. It is always true and mathematicians can prove this.įermat was interested in whole number solutions to Pythagoras’ equation, such that x, y, and z could be any whole number, except zero. Pythagoras’ Theorem is not just a nice idea, or a notion that seems to work for most right-angled triangles. Where z is the length of the hypotenuse, the longest side, and x and y are the lengths of the other two sides. In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares on the other two sides. The page of Arithmetica which inspired Fermat discussed various aspects of Pythagoras’ Theorem, which states that: Pierre de Fermat created the Last Theorem while studying Arithmetica, an ancient Greek text written in about AD 250 by Diophantus of Alexandria.

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. 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. It dominated my own life for four years, because I made a TV documentary, wrote a book and then lectured on the subject. Fermat’s Last Theorem is the most notorious problem in the history of mathematics and surrounding it is one of the greatest stories imaginable.

Fermat's Enigma by Simon Singh