So I have found the following modular solution.

Given: x

^{3}- Dy

^{3}= F

I can solve modulo N, where N is a cubic residue of D, m is the cubed residue, and r is a residue modulo N, where its modular inverse exists:

x = my + r mod N

and

3(2my+r)

^{2}= 4Fr

^{-1 }- r

^{2}mod N

If you wish to see it used, you can just check:

somemath.blogspot.com/2012/05/cubic-diophantine-computations.html

The ideas I use can be generalized further. But I think I went just to a cubic just to see what it might look like. Result seems very pure to me as in I'm not sure how useful it is.

James Harris