A hexagonal tile with number $1$ is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering the tiles $2$ to $7$ in an anti-clockwise direction.

New rings are added in the same fashion, with the next rings being numbered $8$ to $19$, $20$ to $37$, $38$ to $61$, and so on. The diagram below shows the first three rings.

By finding the difference between tile $n$ and each of its six neighbours we shall define $\mathrm{PD}(n)$ to be the number of those differences which are prime.

For example, working clockwise around tile $8$ the differences are $12,29,11,6,1$, and $13$. So $\mathrm{PD}(8)=3$.

In the same way, the differences around tile $17$ are $1,17,16,1,11$, and $10$, hence $\mathrm{PD}(17)=2$.

It can be shown that the maximum value of $\mathrm{PD}(n)$ is $3$.

If all of the tiles for which $\mathrm{PD}(n)=3$ are listed in ascending order to form a sequence, the $10$th tile would be $271$.

Find the $2000$th tile in this sequence.