333,667 is Unique Period Prime with Period 9

Theorem
The only prime number whose reciprocal has a period of $9$ is $333 \, 667$:


 * $\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$

Proof
By long division:

0.000002997000002...     - 333667)1.000000000000000000         667334         3326660         3003003         ---          3236570          3003003          ---           2335670           2335669           ---                 1000000                  667334                 ---                  ......