Fermat's Enigma

by Simon Singh

In order to prove that could not be written as a fraction Euclid used reductio ad absurdum and began by assuming that it could be written as a fraction. He then demonstrated that this hypothetical fraction could be simplified. (Simplification of a fraction means, for example, that the fraction can be simplified to by dividing top and bottom by 2. In turn can be simplified to , which cannot be simplified any further and therefore the fraction is then said to be in its simplest form.) Furthermore, Euclid showed that his simplified fraction, which still was supposed to represent , could be simplified not just once but over and over again an infinite number of times without ever reducing to its simplest form. This is absurd because all fractions must eventually have a simplest form, and therefore the hypothetical fraction cannot exist. Therefore cannot be written as a fraction and is irrational.