Describe solutions to a Diophantine Equation

Question

What does it mean to describe all the solutions to a Diophantine equation? Is it the same as simply finding the solution/general solution?

Answer 1

I guess that describing all the solutions is to put a general equation to easily find them. The following fact first characterizes whether a given Diophantine equation is solvable or not.

For any Diophantine equation of the form $ax+by=c$, it is solvable if and only if $gcd(a,b)$ divides $c$.

If that holds on a given equation, then you have to find at least one solution. Usually, guessing will work. However, if it does not, you can use the extended euclidean algorithm to do so. After finding a solution ($x_0$,$y_0$), the general form for all solutions can be derived as follows:

$$x = x_0 +\frac{b}{g}t, \qquad g = gcd(a,b)$$ $$y = y_0 -\frac{a}{g}t, \qquad g = gcd(a,b)$$