# 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 ------- ......

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:

- $p \mid 10^9 - 1$

Consider:

\(\displaystyle 10^9 - 1\) | \(=\) | \(\displaystyle 999 \, 999 \, 999\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 3^4 \times 37 \times 333 \, 667\) | prime factorization |

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 $333 \, 667$.

$\blacksquare$

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $333,667$