puzzle - How to attack this diophantine equation in 3 variables?

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