Fermat's Last Theorem by Simon Singh Read Free Book Online

– Bell’s book described the existence of a mathematical monster.
    In Pythagoras’ equation the three numbers,
x
,
y
and
z
, are all squared (i.e.
x
2 =
x
×
x
):

    However, the book described a sister equation in which
x
,
y
and
z
are all cubed (i.e.
x 3 = x
×
x
×
x
). The so-called power of
x
in this equation is no longer 2, but rather 3:

    Finding whole number solutions, i.e. Pythagorean triples, to the original equation was relatively easy, but changing the power from ‘2’ to ‘3’ (the square to a cube) and finding whole number solutions to the sister equation appears to be impossible. Generations of mathematicians scribbling on notepads have failed to find numbers which fit the equation perfectly.

    Figure 4. Is it possible to add the building blogs from one cube to another cube, to form a third, larger cube? In this case a 6 × 6 × 6 cube added to an 8 × 8 × 8 cube does not have quite enough building blocks to form a 9 × 9 × 9 cube. There are 216 (6 3 ) building blocks in the first cube, and 512 (8 3 ) in the second. The total is 728 building blogs, which is 1 short of 9 3 .
    With the original ‘squared’ equation, the challenge was to rearrange the tiles in two squares to form a third, larger square. The ‘cubed’ version of the challenge is to rearrange two cubes made of building blocks, to form a third, larger cube. Apparently, no matter what cubes are chosen to begin with, when they are combined the result is either a complete cube with some extra blocks left over, or an incomplete cube. The nearest that anyone has come to a perfect rearrangement is one in which there is one building block too many or too few. For example, if we begin with the cubes 6 3 (
x
3 ) and 8 3 (
y
3 ) and rearrange the building blocks, then we are only one short of making a complete 9 × 9 × 9 cube, as shown in Figure 4 .
    Finding three numbers which fit the cubed equation perfectlyseems to be impossible. That is to say, there appear to be no whole number solutions to the equation

    Furthermore, if the power is changed from 3 (cubed) to any higher number
n
(i.e. 4, 5, 6, …), then finding a solution still seems to be impossible. There appear to be no whole number solutions to the more general equation

    By merely changing the 2 in Pythagoras’ equation to any higher number, finding whole number solutions turns from being relatively simple to being mind-bogglingly difficult. In fact, the great seventeenth-century Frenchman Pierre de Fermat made the astonishing claim that the reason why nobody could find any solutions was that no solutions existed.
    Fermat was one of the most brilliant and intriguing mathematicians in history. He could not have checked the infinity of numbers, but he was absolutely sure that no combination existed which would fit the equation perfectly because his claim was based on proof. Like Pythagoras, who did not have to check every triangle to demonstrate the validity of his theorem, Fermat did not have to check every number to show the validity of his theorem. Fermat’s Last Theorem, as it is known, stated that

    has no whole number solutions for
n
greater than 2.
    As Wiles read each chapter of Bell’s book, he learnt how Fermat had become fascinated by Pythagoras’ work and had eventually come to study the perverted form of Pythagoras’ equation. He then read how Fermat had claimed that even if all the mathematiciansin the world spent eternity looking for a solution to the equation they would fail to find one. He must have eagerly turned the pages, relishing the thought of examining the proof of Fermat’s Last Theorem. However, the proof was not there. It was not anywhere. Bell ended the book by stating that the proof had been lost long ago. There was no hint of what it might have been, no clues as to the proof’s construction or derivation. Wiles found himself puzzled, infuriated and intrigued. He was in