How to attack this diophantine equation in 3 variables? From link:

Find integers a, b and c such that:
987654321a + 123456789b + c = (a + b +
  c)³

 A: Note that x^3 - x = (x-1)x(x+1).  Now let x = (a+b+c) 
and rewrite the equation as
(x-1)x(x+1) = 987654320*a + 123456788*b.
Let D be gcd(987654320,123456788) = 16. There are 
integers A, B so that 987654320*A + 123456788*B = D,
e.g (1, -8).  Pick your favorite x so that x^3 - x is a 
multiple of D, say kD, let a = kA and b = kB, and then 
set c to (x - (a+b)).  If you need a and b to be 
positive, choose x large enough so that kD is big 
enough so that you can subtract multiples of 
987654320*123456788/D^2 from, say ka and add them to kb.
If you need c to be positive as well, then pick x not 
too large, as RHS < 10^9 * (a+b) < 10^9 x, so x larger 
than 10^5 will not get you positive values of c.
Gerhard "Ask Me About System Design" Paseman, 2010.01.12
A: Well, the numbers cannot be more then something like 10^5, so a simple program that cycles through all possible a and b and sees if there's a corresponding c might give you an answer pretty fast -- I dunno, from a minute to a day depending on your computer and programming language.
This also depends on whether you also implement some trivial checks, like try it modulo 2, 3 and 5 first.
