333,667 is Only Prime whose Reciprocal is of 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
-------
......