Compute the number of ordered triples of integers (x, y, z), 1729 < x, y, z < 1999 which satisfy: x^2 + xy + y^2 = y^3 - x^3 and yz + 1 = y^2 + z.
+129849 y^3 - x^3 = (y-x)(y^2 + xy + x^2)
So
(y - x) ( y^2 + xy + x^2) = (y^2 + xy + x^2)
(y - x) = 1
y = x + 1
And
yz + 1 = y^2 + z
(x + 1)z + 1 = (x+1)^2 + z
xz + z + 1 = x^2 + 2x + 1 + z
x^2 + 2x - xz = 0
x ( x + 2 - z) = 0
x = 0 or x + 2 = z
So
{x , y , z} = { x, x + 1, x + 2 }
The number of ordered triples = {1730, 1731, 1732} .....( 1996, 1997, 1998} = 1996 - 1730 + 1 = 267
