Let the plane $$ax + by + cz + d = 0$$ bisect the line joining the points $$(4, -3, 1)$$ and $$(2, 3, -5)$$ at the right angles. If $$a, b, c, d$$ are integers, then the minimum value of $$(a^2 + b^2 + c^2 + d^2)$$ is ________.

The plane bisects the line joining $$(4, -3, 1)$$ and $$(2, 3, -5)$$ at right angles. This means the plane passes through the midpoint and is perpendicular to the line joining these points.

The midpoint is $$\left(\frac{4+2}{2}, \frac{-3+3}{2}, \frac{1-5}{2}\right) = (3, 0, -2)$$.

The direction ratios of the line joining the two points are $$(2-4, 3-(-3), -5-1) = (-2, 6, -6)$$. We can simplify by dividing by $$-2$$: $$(1, -3, 3)$$.

Since the plane is perpendicular to this line, its normal vector has direction ratios $$(1, -3, 3)$$. So the equation of the plane is $$1(x-3) + (-3)(y-0) + 3(z-(-2)) = 0$$.

Expanding: $$x - 3 - 3y + 3z + 6 = 0$$, which gives $$x - 3y + 3z + 3 = 0$$.

So $$a = 1$$, $$b = -3$$, $$c = 3$$, $$d = 3$$.

We verify these are integers with no common factor ($$\gcd(1,3,3,3) = 1$$), so this is already the simplest form.

Therefore $$a^2 + b^2 + c^2 + d^2 = 1 + 9 + 9 + 9 = 28$$.