The Infinite Descent Method
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Lecture Notes

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:

- Assuming $x$, $y$, $\cdots$ is the smallest positive integer solution
- Transforming the given equation and find a smaller set of positive integer solution
- Hence there is a contradiction which means no solution exists

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.