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                 ---                  ......

It remains to be shown that $333 \, 667$ is the only prime number with the required property.

We have that:
 * $333 \, 667 \nmid 10$

From Period of Reciprocal of Prime, the period of such a prime is the order of $10$ modulo $p$.

That is, the smallest integer $d$ such that:
 * $10^d \equiv 1 \pmod p$

From the above long division we know that the period of $\dfrac 1 {333 \, 667}$ is $9$, so $10^9 \equiv 1 \pmod {333 \, 667}$.

The only other possible primes $p$ whose reciprocals might have a period of $9$ must also satisfy:
 * $10^9 \equiv 1 \pmod p$

that is:
 * $p \divides 10^9 - 1$

Consider:

Therefore the only other possible primes whose reciprocals might have a period of $9$ are $3$ and $37$.

From Period of Reciprocal of 37 has Length 3:
 * $\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

and trivially:
 * $\dfrac 1 3 = 0 \cdotp \dot 3$

which has a period of $1$.

As required, the only prime number whose reciprocal has a period of $9$ is $333 \, 667$.