Log of arithmetic progression

Since

$1000 ^ {log (x)} + 1000 ^ {log (y)} + 1000 ^ {log (z)} = 1000 ^ {log (v)}$

Therefore :

$10 ^ {3log (x)} + 10 ^ {3log (y)} + 10 ^ {3log (z)} = 10 ^ {3log (v)}$

$10 ^ {log (x^{3})} + 10 ^ {log (y^{3})} + 10 ^ {log (z^{3})} = 10 ^ {log (v^{3})}$

$x^{3} + y^{3} + z^{3} = v^{3}$

Since x, y, z and v are in an arithmetic progression let :

x = a - d ; y = a ; z = a + d and v = a + 2d

We have:

$(a - d)^{3} + a^{3} + (a + d)^{3} = (a + 2d)^{3}$

Developping and symplifying we get:

$a^{3} - 3a^{2}d - 3ad^{2} - 4d^{3} = 0$

Let $\frac{a}{d} = b$ or a = bd we get:

$b^{3}d^{3} - 3b^{2}d^{3} - 3bd^{3} - 4d^{3} = 0$

Simplifying by $d^{3}$ we get :

$b^{3} - 3b^{2} - 3b - 4 = 0$

$(b - 4) (b^{2} + b + 1) = 0$

The solution to this equation is b = 4 which means a = 4d.

x = 4d - d = 3d, y = a = 4d, z = 4d + d = 5d and v = 4d + 2d = 6d.

The ratio of x : y : z : v = 3 : 4 : 5 : 6.

The minimum possible values are: x = 3, y = 4, z = 5 and v = 6.

The minimum possible values of x + y + z + v = 3 + 4 + 5 + 6 = 18.

$Answer = \boxed{18}$