The most general version of this problem is unsolvable; see "Hilbert's Tenth Problem".
http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem
Specific solutions are generally somewhat difficult.
I believe that some forms of encryption are based on certain Diophantine equations.
At 01:24 PM 12/7/2012, Benito van der Zander wrote:
>Hi,
>is there any way to get all solution of a Diophantine equation symbolically?
>I.e. all solutions of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
>where x,y are unknown integers, and A,B,C,D,E,F are constant terms (mostly numbers, even 0, but some might contain other variables)
>
>There is an long, universal solution algorithm:
>http://www.alpertron.com.ar/METHODS.HTM
>And you can solve them, by looking up the corresponding case there, and putting the terms in the expressions given there,
>but that is boring. And tedious.
>
>Of course the solutions can become quite complex, like (di - E) / B for any di \in divisors(DE - BF).
>But Maxima could calculate DE - BF, and print it, or if it becomes a term without other variables, even directly calculate the divisors.
>(and that DE - BF is actually an easy case)
>
>Benito