### Part (a)
1. Let $k = r - s$.
2. There exist infinitely many $z \in \mathbb{N}$ such that $2 \nmid z^3 + k - 1$. This means that $z^3 + k - 1$ is odd.
3. Define $(x, y) = \left( \frac{z^3 + k}{2}, \frac{z^3 + k - 2}{2} \right)$. Since $z^3 + k$ and $z^3 + k - 2$ are both even, $x$ and $y$ are integers.
4. We have:
$$x^2 + 1 = y^2 + z^3 + k$$
5. Let $n = y^2 + z^3 - s$. Then:
$$s + n = y^2 + z^3$$
and
$$r + n = x^2 + 1^3$$
6. For sufficiently large $z$, $n > 0$.

Thus, there exist infinitely many positive integers $n$ such that both $r + n$ and $s + n$ are bright. $\blacksquare$

### Part (b)
1. Let $k$ be an arbitrary positive integer.
2. Define:
$$m = r^2 s^3 + k^6 s^2 r^3$$
3. Then:
$$sm = (rs^2)^2 + (k^2 sr)^3$$
and
$$rm = (sr^2 k^3)^2 + (rs)^3$$

Thus, there exist infinitely many positive integers $m$ such that both $rm$ and $sm$ are bright. $\blacksquare$