Solving Diophantine Problems

Diophantine problems are word problems that essentially set up algebraic equations to be solved for positive integer or at least rational solutions. They are typically under-determined which opens up the possibility, if not the likelihood, of multiple solutions. This under-determined nature lends an air of mystery to them that adds to their fascination. Solving them seems magical because one intuitively expects them to be impervious to solution.
The category of Diophantine problems includes ...