TITLE: Is there a procedure to solve Diophantine Equations?
QUESTION [6 upvotes]: How would you go about solving a multivariable, non-linear Diophantine Equation?

REPLY [11 votes]: By a famous result of Matiyasevich, there is no universal algorithm which, when fed any Diophantine equation, will determine whether or not that equation has a solution in integers. 
Interestingly, it is still unknown whether there is an algorithm that will always determine whether or not a Diophantine equation has a solution in rationals. 
For quadratic Diophantine equations, in any number of variables, there is an algorithm. For degree $4$ equations, it is known that there is not. The question for cubic equations is unresolved.

REPLY [5 votes]: No, there is no general procedure. This was Hilbert's 10th problem.