## How to attack this diophantine equation in 3 variables?

Find integers a, b and c such that:

987654321a + 123456789b + c = (a + b + c)³

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 The link you provided has solutions on it, together with code for a brute force search. Are you asking for a general algorithm to find integer points on cubic surfaces? – S. Carnahan♦ Oct 12 2009 at 0:57 How are the limits in that program derived? – Cade Roux Oct 12 2009 at 2:36 As far as I can tell, 31427 was chosen because it is the smallest number whose square is larger than 987654321. This covers all cases where a,b, and c are nonnegative. I would not be surprised if there were additional solutions where a and b have opposite signs. – S. Carnahan♦ Oct 12 2009 at 6:09

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.