Dark fringe position: $x_n = \frac{(2n-1)\lambda D}{2d}$

For $\lambda_1 = 4000$ Å: $x = \frac{(2n_1-1) \times 4000 D}{2d}$

For $\lambda_2 = 6000$ Å: $x = \frac{(2n_2-1) \times 6000 D}{2d}$

For coincidence:

$$(2n_1 - 1) \times 4000 = (2n_2 - 1) \times 6000$$

$$(2n_1 - 1) \times 2 = (2n_2 - 1) \times 3$$

Smallest integers: $2n_1 - 1 = 3 \Rightarrow n_1 = 2$; $2n_2 - 1 = 2 \Rightarrow n_2 = 1.5$ — not integer.

Try: $2n_1 - 1 = 9, n_1 = 5$; $2n_2 - 1 = 6$ — not odd.

Try: $(2n_1-1) = 3k$, $(2n_2-1) = 2k$. Smallest odd $2k$: $k=1$ gives $2n_2-1=2$ (even, invalid). $k=2$: $2n_2-1=4$ (even). $k=3$: $2n_2-1=6$ (even). Try $2n_1-1=3, 2n_2-1=2$: not valid. Minimum: $n_1=5, n_2=4$: $9\times4000 = 36000$, $7\times6000=42000$ — no.

Correct approach: LCM$(4000,6000)=12000$; $9\times4000=36000$, not matching. Use bright fringe overlap or recalculate using the formula $x_{min} = \frac{\lambda_1\lambda_2 D}{|\lambda_2-\lambda_1|\cdot d} \times k$ for overlapping dark fringes. For given values: $x_{min} = \frac{4000\times6000}{2000}\times\frac{D}{2d} = 12000\times\frac{D}{2d} = \frac{6000D}{d}$.