The infinite descent method is usually used to prove an indeterminate equation is insolvable in positive integers under certain condition. It relies on the fact that there is a floor (i.e. smallest positive integer). This method is usually employed in the following fashion:
The smallest solution can be typically defined as the solution with the smallest $x$. (Because there may exist situations where $x_1 < x_2$, but $y_1 > y_2$.) Sometimes, it is also possible to define the smallest solution as the one with smallest sum of all the variables. The appropriate definition is determined by the way to derive the contradiction.