The MOD Method
Difficult
Video tutorial
Lecture Notes
The MOD method is an advanced technique to solve indeterminate equation. Using this method requires the ability to observe and discover the modular relationship between the to-be-solved variables. Quite often, this method is used to show there is no solution to the given equation under certain condition. To achieve this goal, a common way is to show the two sides of the equation are not congruent to a carefully chosen modulo.
Examples
Show that if there exist integer $x$, $y$, and $z$ such that $3^x + 4^y=5^z$, then both $x$ and $z$ must be even.(4185)