Let x = (a+b+c)

Then 987654321*a + 123456789*b + (x-a-b) = x^3

Then 987654320*a + 123456788*b = x^3 - x = (x-1)x(x+1)

The GCD of 987654320 and 123456788 is 4, so it appears that the equation has integer solutions whenever (x-1)x(x+1) is a multiple of 4. Three consecutive integers are a multiple of 4 whenever the middle integer is odd or a multiple of 4. In other words, the equation has a solution whenever x <> 2 (mod 4). There are an infinite number of solutions, even after subtracting out that small fraction of them where a or b or c = 0.

*Edited on ***June 13, 2013, 10:49 pm**