For example, solve .
It turns out this equation has no solution - proving that is a lot more complicated.
It is a method of arriving at a contradiction. Say we want to show the equation above has no solution. Thus, we assume that there exists a solution, for example some which makes that equation solvable. And then having that assumption we show a smaller solution can be constructed from the one we had. This leads to a contradiction. Because if we repeat the argument we can obtain even a smaller solution from . And we can do this indefinitely. Since the positive integers cannot be indefinitely decreased our initial assumption must have been wrong.And what is Fermat's method of infinite descent?
That takes too much time. It is almost as bad as asking a person if we can write a book on some topic. What do you think the response will be?EDIT. Can someone on MHF start a tutorial on Number theory? It would be very nice.