Fermat's method of descent

Fermat's method of descent is a way to prove that certain equations have no solutions. It works by assuming a solution exists and then showing you can find a smaller, similar solution. Repeating this process leads to infinitely smaller solutions, which is impossible unless the initial solution was invalid. This contradiction confirms there are no solutions. Essentially, it’s like descending an infinite staircase: if you can always go down one step, you can't start at the top, so your initial assumption must be false. This technique is especially useful in proving that some equations have no integer solutions.